Consider the case of a smooth OKOKO_(K)\mathcal{O}_{K}OK-scheme XXXXX as before. Bhatt-Morrow-Scholze suggest in [18, REMARK 1.16] that ZpBMS/pn(r)XZpBMS/pn(r)XZ_(p)^(BMS)//p^(n)(r)_(X)\mathbb{Z}_{p}^{\mathrm{BMS}} / p^{n}(r)_{X}ZpBMS/pn(r)X should be Schneider's sheaf Sn(r)Sn(r)S_(n)(r)S_{n}(r)Sn(r), and by passage to the limit, there should be a distinguished triangle
This has been proven in a work-in-progress by Bhargav Bhatt and Akhil Mathew [16]. They construct an isomorphism of a version of ZpBMS/pn(r)XZpBMS/pn(r)XZ_(p)^(BMS)//p^(n)(r)_(X)\mathbb{Z}_{p}^{\mathrm{BMS}} / p^{n}(r)_{X}ZpBMS/pn(r)X with Sato's sheaf Tn(r)XTn(r)XT_(n)(r)_(X)\mathfrak{T}_{n}(r)_{X}Tn(r)X in the semi-stable case; using Zhong's extension of Geisser's results, this gives an isomorphism
I am not aware of a categorical framework for the tower FilnTC(A~;Zp)Filnâ¡TCA~;ZpFil^(n)TC(( tilde(A));Z_(p))\operatorname{Fil}^{n} \mathrm{TC}\left(\tilde{A} ; \mathbb{Z}_{p}\right)Filnâ¡TC(A~;Zp) and its layers ZpBMS(r)ZpBMS(r)Z_(p)^(BMS)(r)\mathbb{Z}_{p}^{\mathrm{BMS}}(r)ZpBMS(r), analogous to the framework for Voevodsky's slice tower for KKKKK-theory given by SH(k)SH(k)SH(k)\mathrm{SH}(k)SH(k). As A1A1A^(1)\mathbb{A}^{1}A1-homotopy invariance fails for these theories, one would need a stable homotopy theory with a weaker invariance property, perhaps modeled on the one of the categories of motives with modulus discussed in the previous section, for these theories to find a home, in which the Bhatt-Morrow-Scholze tower (6.5) would be seen as a parallel to Voevodsky's slice tower.
FUNDING
The author gratefully acknowledges support from the DFG through the SPP 1786, and through a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 832833).
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In 1999, Khovanov showed that a link invariant known as the Jones polynomial is the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups, and to explain their meaning - what are they homologies of?
Homological mirror symmetry, formulated by Kontsevich in 1994, naturally produces hosts of homological invariants. Typically though, it leads to invariants which have no particular interest outside of the problem at hand.
I showed recently that there is a new family of mirror pairs of manifolds, for which homological mirror symmetry does lead to interesting invariants and solves the knot categorification problem. The resulting invariants are computable explicitly for any simple Lie algebra, and certain Lie superalgebras.
There are many beautiful strands that connect mathematics and physics. Two of the most fruitful ones are knot theory and mirror symmetry. I will describe a new connection between them. We will find a solution to the knot categorification problem as a new application of homological mirror symmetry.
1.1. Quantum link invariants
In 1984, Jones constructed a polynomial invariant of links in R3R3R^(3)\mathbb{R}^{3}R3 [42]. The Jones polynomial is defined by picking a projection of the link to a plane, the skein relation it satisfies
where n=2n=2n=2n=2n=2, and the value for the unknot. It has the same flavor as the Alexander polynomial, dating back to 1928 [8], which one gets by setting n=0n=0n=0n=0n=0 instead.
The proper framework for these invariants was provided by Witten in 1988, who showed that they originate from three-dimensional Chern-Simons theory based on a Lie algebra LgLg^(L)g{ }^{L} \mathrm{~g}L g [82]. In particular, the Jones polynomial comes from Lg=su2Lg=su2^(L)g=su_(2){ }^{L} \mathrm{~g}=\mathfrak{s u}_{2}L g=su2 with links colored by the defining two-dimensional representation. The Alexander polynomial comes from the same setting by taking LgLg^(L)g{ }^{L} \mathrm{~g}L g to be a Lie superalgebra gll1∣1gll1∣1gll_(1∣1)\mathfrak{g l} \mathfrak{l}_{1 \mid 1}gll1∣1. The resulting link invariants are known as the Uq(Lg)UqLgU_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) quantum group invariants. The relation to quantum groups was discovered by Reshetikhin and Turaev [67].
1.2. The knot categorification problem
The quantum invariants of links are Laurant polynomials in q1/2q1/2q^(1//2)q^{1 / 2}q1/2, with integer coefficients. In 1999, Khovanov showed [48, 49] that one can associate to a projection of the link to a plane a bigraded complex of vector spaces
are independent of the choice of projection; they are themselves link invariants.
1.2.1.
Khovanov's construction is part of the categorification program initiated by Crane and Frenkel [25], which aims to lift integers to vector spaces and vector spaces to categories.
A toy model of categorification comes from a Riemannian manifold MMMMM, whose Euler characteristic
is categorified by the cohomology Hk(M)=kerdk/imdk−1Hk(M)=kerâ¡dk/imâ¡dk−1H^(k)(M)=ker d_(k)//im d_(k-1)\mathscr{H}^{k}(M)=\operatorname{ker} d_{k} / \operatorname{im} d_{k-1}Hk(M)=kerâ¡dk/imâ¡dk−1 of the de Rham complex
The Euler characteristic is, from the physics perspective, the partition function of supersymmetric quantum mechanics with MMMMM as a target space χ(M)=Tr(−1)Fe−βHχ(M)=Trâ¡(−1)Fe−βHchi(M)=Tr(-1)^(F)e^(-beta H)\chi(M)=\operatorname{Tr}(-1)^{F} e^{-\beta H}χ(M)=Trâ¡(−1)Fe−βH, with Laplacian H=dd∗+d∗dH=dd∗+d∗dH=dd^(**)+d^(**)dH=d d^{*}+d^{*} dH=dd∗+d∗d as the Hamiltonian, and d=∑kdkd=∑k dkd=sum_(k)d_(k)d=\sum_{k} d_{k}d=∑kdk as the supersymmetry operator. If hhhhh is a Morse function on MMMMM, the complex can be replaced by a Morse-Smale-Witten complex Ch∗Ch∗C_(h)^(**)C_{h}^{*}Ch∗ with the differential dh=ehde−hdh=ehde−hd_(h)=e^(h)de^(-h)d_{h}=e^{h} d e^{-h}dh=ehde−h. The complex Ch∗Ch∗C_(h)^(**)C_{h}^{*}Ch∗ is the space of perturbative ground states of a σσsigma\sigmaσ-model on MMMMM with potential hhhhh [81]. The action of the differential dhdhd_(h)d_{h}dh is generated by solutions to flow equations, called instantons.
1.2.2.
Khovanov's remarkable categorification of the Jones polynomial is explicit and easily computable. It has generalizations of similar flavor for Lg=unLg=un^(L)g=u_(n){ }^{L} \mathfrak{g}=\mathfrak{} u_{n}Lg=un, and links colored by its minuscule representations [51].
In 2013, Webster showed [78] that for any LgLg^(L)g{ }^{L} \mathrm{~g}LÂ g, there exists an algebraic framework for categorification of Uq(Lg)UqLgU_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(LÂ g) invariants of links in R3R3R^(3)\mathbb{R}^{3}R3, based on a derived category of modules of an associative algebra. The KLRW algebra, defined in [78], generalizes the algebras of Khovanov and Lauda [50] and Rouquier [68]. Unlike Khovanov's construction, Webster's categorification is anything but explicit.
1.2.3.
Despite the successes of the program, one is missing a fundamental principle which explains why is categorification possible - the construction has no right to exist. Unlike in our toy example of categorification of the Euler characteristic of a Riemanniann manifold, Khovanov's construction and its generalizations did not come from either geometry or physics in any unified way. The problem Khovanov initiated is to find a general framework for link homology, that works uniformly for all Lie algebras, explains what link homology groups are, and why they exist.
1.3. Homological invariants from mirror symmetry
The solution to the problem comes from a new relation between mirror symmetry and representation theory.
Homological mirror symmetry relates pairs of categories of geometric origin [55]: a derived category of coherent sheaves and a version of the derived Fukaya category, in which complementary aspects of the theory are simple to understand. Occasionally, one can make mirror symmetry manifest, by showing that both categories are equivalent to a derived category of modules of a single algebra.
I will describe a new family of mirror pairs, in which homological mirror symmetry can be made manifest and leads to the solution to the knot categorification problem [1, 2]. Many special features exist in this family, in part due to its deep connections to representation theory. As a result, the theory is solvable explicitly, as opposed to only formally [4,5].
1.4. The solution
We will get not one, but two solutions to the knot categorification problem. The first solution [1] is based on DXDXDX\mathscr{D} XDX, the derived category of equivariant coherent sheaves on a certain holomorphic symplectic manifold XXX\mathcal{X}X, which plays a role in the geometric Langlands correspondence. Recently, Webster proved that DXDXD_(X)\mathscr{D}_{X}DX is equivalent to DADAD_(A)\mathscr{D}_{\mathscr{A}}DA, the derived category of modules of an algebra AAA\mathscr{A}A which is a cylindrical version of the KLRW algebra from [79,80]. The generalization allows the theory to describe links in R2×S1R2×S1R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1, as well as in R3R3R^(3)\mathbb{R}^{3}R3.
The second solution [2] is based on DYDYD_(Y)\mathscr{D}_{Y}DY, the derived Fukaya-Seidel category of a certain manifold YYYYY with potential WWWWW. The theory generalizes Heegard-Floer theory [63,64[63,64[63,64[63,64[63,64, 66], which categorifies the Alexander polynomial, from Lg=gl1∣1Lg=gl1∣1^(L)g=gl_(1∣1){ }^{L} \mathrm{~g}=\mathrm{gl}_{1 \mid 1}L g=gl1∣1, to arbitrary LgLg^(L)g{ }^{L} \mathrm{~g}L g.
The two solutions are related by equivariant homological mirror symmetry, which is not an equivalence of categories, but a correspondence of objects and morphisms coming from a pair of adjoint functors. In DXDXDX\mathscr{D} XDX, we will learn which question we need to ask to obtain Uq(Lg)UqLgU_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(LÂ g) link homology. In DYDYD_(Y)\mathscr{D}_{Y}DY, we will learn how to answer it.
In [5], we give an explicit algorithm for computing homological link invariants from DYDYD_(Y)\mathscr{D}_{Y}DY, for any simple Lie algebra LgLg^(L)g{ }^{L} \mathrm{~g}L g and links colored by its minuscule representations. It has an extension to Lie superalgebras Lg=glm∣nLg=glm∣n^(L)g=gl_(m∣n){ }^{L} \mathfrak{g}=\mathfrak{g} \mathfrak{l}_{m \mid n}Lg=glm∣n and spm∣2nspm∣2nsp_(m∣2n)\mathfrak{s} \mathfrak{p}_{m \mid 2 n}spm∣2n. In [4], we set the mathematical foundations of DYDYD_(Y)\mathscr{D}_{Y}DY and prove (equivariant) homological mirror symmetry relating it to DXDXDX\mathscr{D} XDX.
2. KNOT INVARIANTS AND CONFORMAL FIELD THEORY
Most approaches to categorification of Uq(Lg)UqLgU_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(LÂ g) link invariants start with quantum groups and their modules. We will start by recalling how quantum groups came into the story. The seeming detour will help us understand how Uq(Lg)UqLgU_(q)(^(L)g)U_{\mathfrak{q}}\left({ }^{L} \mathfrak{g}\right)Uq(Lg) link invariants arise from geometry, and what categorifies them.
2.1. Knizhnik-Zamolodchikov equation and quantum groups
Chern-Simons theory associates to a punctured Riemann surface AAA\mathscr{A}A a vector space, its Hilbert space. As Witten showed [82], the Hilbert space is finite dimensional, and spanned by vectors that have a name. They are known as conformal blocks of the affine Lie algebra L^gκL^gκwidehat(L)_(g_(kappa))\widehat{L}_{\mathrm{g}_{\kappa}}L^gκ. The effective level κκkappa\kappaκ is an arbitrary complex number, related to qqqqq by q=e2πiκq=e2Ï€iκq=e^((2pi i)/(kappa))q=e^{\frac{2 \pi i}{\kappa}}q=e2Ï€iκ. While in principle arbitrary representations of LgLg^(L)g{ }^{L} \mathrm{~g}L g can occur, in relating to geometry and categorification, we will take them to be minuscule.
To get invariants of knots in R3R3R^(3)\mathbb{R}^{3}R3, one typically takes AAA\mathscr{A}A to be a complex plane with punctures. It is equivalent, but for our purposes better, to take AAA\mathscr{A}A to be an infinite complex cylinder. This way, we will be able to describe invariants of links in R2×S1R2×S1R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1, as well.
2.1.1.
Every conformal block, and hence every state in the Hilbert space, can be obtained explicitly as a solution to a linear differential equation discovered by Knizhnik and Zamolodchikov in 1984 [53]. The KZ equation we get is of trigonometric type, schematically
since AAA\mathscr{A}A is an infinite cylinder. Here, ∂i=ai∂∂ai∂i=ai∂∂aidel_(i)=a_(i)(del)/(dela_(i))\partial_{i}=a_{i} \frac{\partial}{\partial a_{i}}∂i=ai∂∂ai, where aiaia_(i)a_{i}ai is any of the nnnnn punctures in the interior of AAA\mathcal{A}A, colored by a representation ViViV_(i)V_{i}Vi of LgLg^(L)g{ }^{L} \mathrm{~g}L g. The right hand side of (2.1) is given in terms of classical rrrrr-matrices of LgLg^(L)g{ }^{L} \mathrm{~g}L g, and acts irreducibly on a subspace of V1⊗⋯⊗VnV1⊗⋯⊗VnV_(1)ox cdots oxV_(n)V_{1} \otimes \cdots \otimes V_{n}V1⊗⋯⊗Vn of a fixed weight ννnu\nuν, where VVV\mathcal{V}V takes values [32,33].
The KZKZKZ\mathrm{KZ}KZ equations define a flat connection on a vector bundle over the configuration space of distinct points a1,…,an∈Aa1,…,an∈Aa_(1),dots,a_(n)inAa_{1}, \ldots, a_{n} \in \mathcal{A}a1,…,an∈A. The flatness of the connection is the integrability condition for the equation.
2.1.2.
The monodromy problem of the KZKZKZ\mathrm{KZ}KZ equation, which is to describe analytic continuation of its fundamental solution along a path in the configuration space, has an explicit solution. Drinfeld [30] and Kazhdan and Lustig [47] proved that that the monodromy matrix BBB\mathfrak{B}B of the KZKZKZ\mathrm{KZ}KZ connection is a product of RRRRR-matrices of the Uq(Lg)UqLgU_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(LÂ g) quantum group corresponding to LgLg^(L)g{ }^{L} \mathrm{~g}LÂ g. The RRRRR-matrices describe reorderings of neighboring pairs of punctures.
The monodromy matrix BBB\mathfrak{B}B is the Chern-Simons path integral on A×[0,1]A×[0,1]Axx[0,1]\mathscr{A} \times[0,1]A×[0,1] in presence of a colored braid. By composing braids, we get a representation of the affine braid group based on the Uq(Lg)UqLgU_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) quantum group, acting on the space of solutions to the KZKZKZ\mathrm{KZ}KZ equation. The braid group is affine, since AAA\mathscr{A}A is a cylinder and not a plane.
2.1.3.
Any link can be represented as a plat closure of some braid. The Chern-Simons path integral together with the link computes a very specific matrix element of the braiding matrix BBB\mathfrak{B}B, picked out by a pair of states in the Hilbert space corresponding to the top and the bottom of Figure 1 .
FIGURE 1
Every link arises as a plat closure of a braid.
These states, describing a collection of cups and caps, are very special solutions of the KZKZKZ\mathrm{KZ}KZ equation in which pairs of punctures, colored by conjugate representations ViViV_(i)V_{i}Vi and Vi∗Vi∗V_(i)^(**)V_{i}^{*}Vi∗, come together and fuse to disappear. In this way, both fusion and braiding enter the problem.
2.2. A categorification wishlist
To categorify Uq(Lg)UqLgU_(q)(^(L)(g))U_{\mathfrak{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) invariants of links in R3R3R^(3)\mathbb{R}^{3}R3, we would like to associate, to the space of conformal blocks of LG^LG^widehat(L_(G))\widehat{L_{\mathcal{G}}}LG^ on the Riemann surface AAA\mathcal{A}A, a bigraded category, which in addition to the cohomological grading has a grading associated to qqq\mathrm{q}q. Additional rk(Lg)rkLgrk(^(L)(g))\mathrm{rk}\left({ }^{L} \mathrm{~g}\right)rk(L g) gradings are needed to categorify invariants of links in R2×S1R2×S1R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1, as they depend on the choice of a flat LgLg^(L)g{ }^{L} \mathrm{~g}L g connection around the S1S1S^(1)S^{1}S1. To braids, we would like to associate functors between the categories corresponding to the top and bottom. To links, we would like to associate a vector space whose elements are morphisms between the objects of the categories associated to the top and bottom, up to the action of the braiding functor. Moreover, we would like to do this in a way that recovers quantum link invariants upon decategorification. One typically proceeds by coming up with a category, and then works to prove that decategorification gives the link invariants one set out to categorify. A virtue of the solutions in [1,2][1,2][1,2][1,2][1,2] is that the second step is automatic.
3. MIRROR SYMMETRY
Mirror symmetry is a string duality which relates σσsigma\sigmaσ-models on a pair of Calabi-Yau manifolds XXX\mathcal{X}X and yyy\mathscr{y}y. Its mathematical imprint are relations between very different problems in complex geometry of XXX\mathcal{X}X ("B-type") and symplectic geometry of yyyyy ("A-type"), and vice versa.
Mirror symmetry was discovered as a duality of σσsigma\sigmaσ-models on closed Riemann surfaces DDDDD. In string theory, one must allow Riemann surfaces with boundaries. This enriches the theory by introducing "branes," which are boundary conditions at ∂D∂Ddel D\partial D∂D and naturally objects of a category [9].
By asking how mirror symmetry acts on branes turned out to yield deep insights into mirror symmetry. One such insight is due to Strominger, Yau, and Zaslov [75], who showed that in order for every point-like brane on XXX\mathcal{X}X to have a mirror on YYYYY, mirror manifolds have to be fibered by a pair of (special Lagrangian) dual tori TTTTT and T∨T∨T^(vv)T^{\vee}T∨, over a common base.
3.1. Homological mirror symmetry
Kontsevich conjectured in his 1994 ICM address [55] that mirror symmetry should be understood as an equivalence of a pair of categories of branes, one associated to complex geometry of XXX\mathcal{X}X, the other to symplectic geometry of yyy\mathscr{y}y.
The category of branes associated to complex geometry of XXX\mathcal{X}X is the derived category of coherent sheaves,
Its objects are "B-type branes," supported on complex submanifolds of XXX\mathcal{X}X. The category of branes associated to symplectic geometry is the derived Fukaya category
whose objects are "A-type branes," supported on Lagrangian submanifolds of yyyyy, together with a choice of orientation and a flat bundle. For example, mirror symmetry should map the structure sheaf of a point in XXX\mathcal{X}X to a Lagrangian brane in yyyyy supported on a T∨T∨T^(vv)T^{\vee}T∨ fiber. The choice of a flat U(1)U(1)U(1)U(1)U(1) connection is the position of the point in the dual fiber TTTTT.
Kontsevich' homological mirror symmetry is a conjecture that the category of Bbranes on XXX\mathcal{X}X and the category of A-branes on yyyyy are equivalent,
Dx≅DyDx≅DyDx~=Dy\mathscr{D} x \cong \mathscr{D} yDx≅Dy
and that this equivalence characterizes what mirror symmetry is.
3.2. Quantum differential equation and its monodromy
Knizhnik-Zamolodchikov equation, which plays a central role in knot theory, has a geometric counterpart. In the world of mirror symmetry, there is an equally fundamental differential equation,
The equation is known as the quantum differential equation of XXX\mathcal{X}X. Both the equation and its monodromy problem featured prominently, starting with the first papers on mirror symmetry, see [37] for an early account. In (3.1), (Ci)αβ=CγiαδηδβCiαβ=Cγiαδηδβ(C_(i))_(alpha)^(beta)=C_(gamma_(i)alpha delta)eta^(delta beta)\left(C_{i}\right)_{\alpha}^{\beta}=C_{\gamma_{i} \alpha \delta} \eta^{\delta \beta}(Ci)αβ=Cγiαδηδβ is a connection on a vector bundle with fibers Heven (X)=⨁kHk(X,∧kTX∗)Heven (X)=â¨k HkX,∧kTX∗H^("even ")(X)=bigoplus_(k)H^(k)(X,^^^(k)T_(X)^(**))H^{\text {even }}(\mathcal{X})=\bigoplus_{k} H^{k}\left(\mathcal{X}, \wedge^{k} T_{X}^{*}\right)Heven (X)=â¨kHk(X,∧kTX∗) over the complexified Kahler moduli space. The derivative stands for ∂i=ai∂∂ai∂i=ai∂∂aidel_(i)=a_(i)(del)/(dela_(i))\partial_{i}=a_{i} \frac{\partial}{\partial a_{i}}∂i=ai∂∂ai, so that ∂iad=(γi,d)ad∂iad=γi,daddel_(i)a^(d)=(gamma_(i),d)a^(d)\partial_{i} a^{d}=\left(\gamma_{i}, d\right) a^{d}∂iad=(γi,d)ad for a curve of degree d∈H2(X)d∈H2(X)d inH_(2)(X)d \in H_{2}(\mathcal{X})d∈H2(X). The connection comes from quantum multiplication with classes γi∈H2(X)γi∈H2(X)gamma_(i)inH^(2)(X)\gamma_{i} \in H^{2}(\mathcal{X})γi∈H2(X). Given three de Rham cohomology classes on XXX\mathcal{X}X, their quantum product
From the mirror perspective of yyyyy, the connection is the classical Gauss-Manin connection on the vector bundle over the moduli space of complex structures on yyyyy, with fibers the mid-dimensional cohomology Hd(y)Hd(y)H^(d)(y)H^{d}(y)Hd(y) as mirror symmetry identifies Hk(X,∧kTX∗)HkX,∧kTX∗H^(k)(X,^^^(k)T_(X)^(**))H^{k}\left(\mathcal{X}, \wedge^{k} T_{X}^{*}\right)Hk(X,∧kTX∗) with Hk(y,∧d−kTy∗)Hky,∧d−kTy∗H^(k)(y,^^^(d-k)T_(y)^(**))H^{k}\left(y, \wedge^{d-k} T_{y}^{*}\right)Hk(y,∧d−kTy∗).
3.2.1.
Solutions to the equation live in a vector space, spanned by K-theory classes of branes [22,36,41,46]. These are B-type branes on XXX\mathcal{X}X, objects of DXDXD_(X)\mathscr{D}_{X}DX, and A-type branes on YYYYY, objects of DyDyDy\mathscr{D} yDy. A characteristic feature is that the equation and its solutions mix the A- and B-type structures on the same manifold.
From the perspective of XXX\mathcal{X}X, the solutions of the quantum differential equation come from Gromov-Witten theory. They are obtained by counting holomorphic maps from a domain curve DDDDD to XXX\mathcal{X}X, where DDDDD is best thought of as an infinite cigar [39,40][39,40][39,40][39,40][39,40] together with insertions of a class in α∈Heven ∗(X)α∈Heven ∗(X)alpha inH_("even ")^(**)(X)\alpha \in H_{\text {even }}^{*}(\mathcal{X})α∈Heven ∗(X) at the origin, and [F]∈K(X)[F]∈K(X)[F]in K(X)[\mathcal{F}] \in K(\mathcal{X})[F]∈K(X) at infinity. The latter is the K-theory class of a B-type brane F∈DxF∈DxFinDx\mathscr{F} \in \mathscr{D} xF∈Dx which serves as the boundary condition at the S1S1S^(1)S^{1}S1 boundary at infinity of DDDDD. In the mirror yyyyy, the A- and B-type structures get exchanged. In the interior of DDDDD, supersymmetry is preserved by B-type twist, and at the boundary at infinity we place an A-type brane L∈DyL∈DyLinDy\mathscr{L} \in \mathscr{D} yL∈Dy, whose KKKKK-theory class picks which solution of the equation we get.
3.2.2.
One of the key mirror symmetry predictions is that monodromy of the quantum differential equation gets categorified by the action of derived autoequivalences of DXDXDX\mathscr{D} XDX. It is related by mirror symmetry to the monodromy of the Gauss-Manin connection, computed by Picard-Lefshetz theory, whose categorification by DyDyDy\mathscr{D} yDy is developed by Seidel [71].
The Knizhnik-Zamolodchikov equation not only has the same flavor as the quantum differential equation, but for some very special choices of XXX\mathcal{X}X, they coincide. For the time being, we will take LgLg^(L)g{ }^{L} \mathrm{~g}LÂ g to be simply laced, so it coincides with its Langlands dual ggg\mathrm{g}g.
4.1. The geometry
The manifold XXX\mathcal{X}X may be described as the moduli space of GGGGG-monopoles on
with prescribed singularities. The monopole group GGGGG is related to LGLG^(L)G{ }^{L} GLG, the Chern-Simons gauge group, by Langlands or electric-magnetic duality. In Chern-Simons theory, the knots are labeled by representations of LGLG^(L)G{ }^{L} GLG and viewed as paths of heavy particles, charged electrically under LGLG^(L)G{ }^{L} GLG. In the geometric description, the same heavy particles appear as singular, Dirac-type monopoles of the Langlands dual group GGGGG. The fact the magnetic description is what is needed to understand categorification was anticipated by Witten [84-87].
4.1.1.
Place a singular GGGGG monopole for every finite puncture on A≅R×S1A≅R×S1A~=RxxS^(1)\mathscr{A} \cong \mathbb{R} \times S^{1}A≅R×S1, at the point on RRR\mathbb{R}R obtained by forgetting the S1S1S^(1)S^{1}S1. Singular monopole charges are elements of the cocharacter lattice of GGGGG, which Langlands duality identifies with the character lattice of LGLG^(L)G{ }^{L} GLG. Pick the charge of the monopole to be the highest weight μiμimu_(i)\mu_{i}μi of the LGLG^(L)G{ }^{L} GLG representation ViViV_(i)V_{i}Vi coloring the puncture. The relative positions of singular monopoles on R3R3R^(3)\mathbb{R}^{3}R3 are the moduli of the metric on XXX\mathcal{X}X, so we will hold them fixed.
The smooth monopole charge is a positive root of LGLG^(L)G{ }^{L} GLG; choose it so that the total monopole charge is the weight ννnu\nuν of subspace of representation ⊗iVi⊗iViox_(i)V_(i)\otimes_{i} V_{i}⊗iVi, where the conformal blocks take values. For our current purpose, it suffices to assume
is a dominant weight; LeaLea^(L)e_(a){ }^{L} e_{a}Lea are the simple positive roots of LgLg^(L)g{ }^{L} \mathrm{~g}L g. Provided μiμimu_(i)\mu_{i}μi are minuscule co-weights of GGGGG and no pairs of singular monopoles coincide, the monopole moduli space XXX\mathcal{X}X is a smooth hyper-Kahler manifold of dimension
It is parameterized, in part, by positions of smooth monopoles on R3R3R^(3)\mathbb{R}^{3}R3.
4.1.2.
A choice of complex structure on XXX\mathcal{X}X reflects a split of R3R3R^(3)\mathbb{R}^{3}R3 as R×CR×CRxxC\mathbb{R} \times \mathbb{C}R×C. The relative positions of singular monopoles on CCC\mathbb{C}C become the complex structure moduli, and the relative positions of monopoles on RRR\mathbb{R}R the Kahler moduli.
This identifies the complexified Kahler moduli space of XXX\mathcal{X}X (where the Kahler form gets complexified by a periodic two-form) with the configuration space of nnnnn distinct punctures on A=R×S1A=R×S1A=RxxS^(1)\mathscr{A}=\mathbb{R} \times S^{1}A=R×S1, modulo overall translations, as in Figure 2 .
FIGURE 2
Punctures on AAA\mathcal{A}A correspond to singular GGGGG-monopoles on R∈R×CR∈R×CRinRxxC\mathbb{R} \in \mathbb{R} \times \mathbb{C}R∈R×C.
4.1.3.
As a hyper-Kahler manifold, XXX\mathcal{X}X has more symmetries than a typical Calabi-Yau. For its quantum cohomology to be nontrivial, and for the quantum differential equation to coincide with the KZKZKZ\mathrm{KZ}KZ equation, we need to work equivariantly with respect to a torus action that scales its holomorphic symplectic form
For this to be a symmetry, we will place all the singular monopoles at the origin of C;XC;XC;X\mathbb{C} ; \mathcal{X}C;X has a larger torus of symmetries
where ΛΛLambda\LambdaΛ preserves the holomorphic symplectic form, and comes from the Cartan torus of GGGGG. The equivariant parameters of the ΛΛLambda\LambdaΛ-action correspond to the choice of a flat LGLG^(L)G{ }^{L} GLG connection of Chern-Simons theory on R2×S1R2×S1R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1.
4.1.4.
The same manifold XXX\mathcal{X}X has appeared in mathematics before, as a resolution of a transversal slice in the affine Grassmannian GrG=G((z))/G[[z]]GrG=G((z))/G[[z]]Gr_(G)=G((z))//G[[z]]\operatorname{Gr}_{G}=G((z)) / G[[z]]GrG=G((z))/G[[z]] of GGGGG, often denoted by
The two are related by thinking of monopole moduli space XXX\mathcal{X}X as obtained by a sequence of Hecke modifications of holomorphic GGGGG-bundles on CCC\mathbb{C}C [45].
Manifold XXX\mathcal{X}X is also the Coulomb branch of a 3d quiver gauge theory with N=4N=4N=4\mathcal{N}=4N=4 supersymmetry, with quiver based on the Dynkin diagram of ggggg, see e.g. [19]. The ranks of the flavor and gauge symmetry groups are determined from the weights μμmu\muμ and ννnu\nuν.
4.1.5.
The vector μ→=(μ1,…,μn)μ→=μ1,…,μnvec(mu)=(mu_(1),dots,mu_(n))\vec{\mu}=\left(\mu_{1}, \ldots, \mu_{n}\right)μ→=(μ1,…,μn) in (4.3) encodes singular monopole charges, and the order in which they appear on RRR\mathbb{R}R, and ννnu\nuν is the total monopole charge. The ordering of entries of μ→μ→vec(mu)\vec{\mu}μ→ is a choice of a chamber in the Kahler moduli. We will suppress μ→μ→vec(mu)\vec{\mu}μ→ for the most part, and denote all the corresponding distinct symplectic manifolds by XXX\mathcal{X}X.
4.1.6.
By a recent theorem of Danilenko [26], the Knizhnik-Zamolodchikov equation corresponding to the Riemann surface A=R×S1A=R×S1A=RxxS^(1)\mathcal{A}=\mathbb{R} \times S^{1}A=R×S1, with punctures colored by minuscule representations ViViV_(i)V_{i}Vi of LgLg^(L)g{ }^{L} \mathrm{~g}L g, coincides with the quantum differential equation of the TTTTT-equivariant Gromov-Witten theory on X=Grμ→νX=Grμ→νX=Gr^( vec(mu))_(nu)\mathcal{X}=\mathrm{Gr}^{\vec{\mu}}{ }_{\nu}X=Grμ→ν.
This has many deep consequences.
4.2. Branes and braiding
Since the KZKZKZ\mathrm{KZ}KZ equation is the quantum-differential equation of TTTTT-equivariant Gromov-Witten theory of XXX\mathcal{X}X, the space of its solutions gets identified with KT(X)KT(X)K_(T)(X)K_{T}(\mathcal{X})KT(X), the TTTTT-equivariant KKKKK-theory of XXX\mathcal{X}X.
This is the KKKKK-group of the category of its B-type branes, the derived category of TTTTT-equivariant coherent sheaves on XXX\mathcal{X}X,
This connection between the KZKZKZ\mathrm{KZ}KZ equation and DxDxDx\mathscr{D} xDx is the starting point for our first solution of the categorification problem.
4.2.1.
A colored braid with nnnnn strands in A×[0,1]A×[0,1]Axx[0,1]\mathcal{A} \times[0,1]A×[0,1] has a geometric interpretation as a path in the complexified Kahler moduli of XXX\mathcal{X}X that avoids singularities, as the order of punctures on AAA\mathcal{A}A corresponds to a choice of chamber in the Kahler moduli of XXX\mathcal{X}X.
The monodromy of the quantum differential equation along this path acts on KT(X)KT(X)K_(T)(X)K_{T}(\mathcal{X})KT(X). Since the quantum differential equation coincides with the KZKZKZ\mathrm{KZ}KZ equation, by the theorem of [26], KT(X)KT(X)K_(T)(X)K_{T}(\mathcal{X})KT(X) becomes a module for Uq(Lg)UqLgU_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g), corresponding to the weight ννnu\nuν subspace of representation V1⊗⋯⊗VnV1⊗⋯⊗VnV_(1)ox cdots oxV_(n)V_{1} \otimes \cdots \otimes V_{n}V1⊗⋯⊗Vn
The fact that derived equivalences of DxDxDx\mathscr{D} xDx categorify this action is not only an expectation, but also a theorem by Bezrukavnikov and Okounkov [14], whose proof makes use of quantization of XXX\mathcal{X}X in characteristic ppppp.
4.2.2.
From physics perspective, the reason derived equivalences of DxDxDx\mathscr{D} xDx had to categorify the action of monodromy of the quantum differential equation on KT(X)KT(X)K_(T)(X)K_{T}(\mathcal{X})KT(X) is as follows.
Braid group acts, in the σσsigma\sigmaσ-model on the cigar DDDDD from Section 3.2.1, by letting the moduli of XXX\mathcal{X}X vary according to the braid near the boundary at infinity. The Euclidean time, running along the cigar, is identified with the time along the braid. This leads to a Berry phase-type problem studied by Cecotti and Vafa [22]. It follows that the σσsigma\sigmaσ-model on the annulus, with moduli that vary according to the braid, computes the matrix element of the monodromy BBB\mathfrak{B}B, picked out by a pair of branes F0F0F_(0)\mathscr{F}_{0}F0 and F1F1F_(1)\mathscr{F}_{1}F1 at its boundaries.
The σσsigma\sigmaσ-model on the same Euclidian annulus, where we take the time to run around S1S1S^(1)S^{1}S1 instead, computes the index of the supercharge QQQQQ preserved by the two branes. The cohomology of QQQQQ is computed by DxDxDx\mathscr{D} xDx as its most basic ingredient, the space of morphisms
between the branes. This is the space of supersymmetric ground states of the σσsigma\sigmaσ-model on a strip, obtained by cutting the annulus open. We took here all the variations of moduli to happen near one boundary, at the expense of changing a boundary condition from F0F0F_(0)\mathscr{F}_{0}F0 to BF0BF0BF_(0)\mathscr{B} \mathcal{F}_{0}BF0. This does not affect the homology [1,35][1,35][1,35][1,35][1,35], for the very same reason the theory depends on the homotopy type of the braid only. Per construction, the graded Euler characteristic of the homology theory, computed by closing the strip back up to the annulus, is the braiding matrix element,
between the conformal blocks V0,1=V[F0,1]V0,1=VF0,1V_(0,1)=V[F_(0,1)]\mathcal{V}_{0,1}=\mathcal{V}\left[\mathcal{F}_{0,1}\right]V0,1=V[F0,1] of the two branes.
Thus, by viewing the same Euclidian annulus in two different ways, we learn that the braid group action on the derived category
manifestly categorifies the monodromy matrix B∈Uq(Lg)B∈UqLgBinU_(q)(^(L)(g))\mathfrak{B} \in U_{\mathfrak{q}}\left({ }^{L} \mathrm{~g}\right)B∈Uq(L g) of the KZKZKZ\mathrm{KZ}KZ equation.
4.3. Link invariants from perverse equivalences
The quantum Uq(Lg)UqLgU_(q)(^(L)(g))U_{\mathcal{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) invariants of knots and links are matrix elements of the braiding matrix BBB\mathscr{B}B, so they too will be categorified by DxDxDx\mathscr{D} xDx, provided we can identify objects U∈DxU∈DxU inDxU \in \mathscr{D} xU∈Dx which serve as cups and caps.
Conformal blocks corresponding to cups and caps are defined using fusion [62]. The geometric analogue of fusion, in terms of XXX\mathcal{X}X and its category of branes, was shown in [1] to be the existence of certain perverse filtrations on DxDxDx\mathscr{D} xDx, defined by abstractly by Chuang and Rouqiuer [24]. The utility of perverse filtrations for understanding the action of braiding on DxDxDx\mathscr{D} xDx parallels the utility of fusion in describing the action of braiding in conformal field theory. In particular, it leads to identification of the cup and cap branes UUU\boldsymbol{U}U we need, and a
4.3.1.
As we bring a pair of punctures at aiaia_(i)a_{i}ai and ajaja_(j)a_{j}aj on AAA\mathcal{A}A together, we get a new natural basis of solutions to the KZKZKZ\mathrm{KZ}KZ equation, called the fusion basis, whose virtue is that it diagonalizes braiding. The possible eigenvectors are labeled by the representations
that occur in the tensor product of representations ViViV_(i)V_{i}Vi and VjVjV_(j)V_{j}Vj labeling the punctures. Because ViViV_(i)V_{i}Vi and VjVjV_(j)V_{j}Vj are minuscule representations, the nonzero multiplicities on the right-hand side are all equal to 1 . The cap arises as a special case, obtained by starting with a pair of conjugate representations ViViV_(i)V_{i}Vi and Vi⋆Vi⋆V_(i)^(***)V_{i}^{\star}Vi⋆, and picking the trivial representation in their tensor product.
4.3.2.
From perspective of XXX\mathcal{X}X, a pair of singular monopoles of charges μiμimu_(i)\mu_{i}μi and μjμjmu_(j)\mu_{j}μj are coming together on RRR\mathbb{R}R, as in Figure 2, and we approach a wall in Kahler moduli at which XXX\mathcal{X}X develops a singularity. At the singularity, a collection of cycles vanishes. This is due to monopole bubbling phenomena described by Kapustin and Witten in [45].
The types of monopole bubbling that can occur are labeled by representations VkmVkmV_(k_(m))V_{k_{m}}Vkm that occur in the tensor product Vi⊗VjVi⊗VjV_(i)oxV_(j)V_{i} \otimes V_{j}Vi⊗Vj. The moduli space of monopoles whose positions we need to tune for the bubbling of type VkmVkmV_(k_(m))V_{k_{m}}Vkm to occur is Grμkm(μi,μj)=T∗FkmGrμkmμi,μj=T∗FkmGr_(mu_(k_(m)))^((mu_(i),mu_(j)))=T^(**)F_(k_(m))\operatorname{Gr}_{\mu_{k_{m}}}^{\left(\mu_{i}, \mu_{j}\right)}=T^{*} F_{k_{m}}Grμkm(μi,μj)=T∗Fkm, where μkmμkmmu_(k_(m))\mu_{k_{m}}μkm is the highest weight of VkmVkmV_(k_(m))V_{k_{m}}Vkm. This space is transverse to the locus where exactly μi+μj−μkmμi+μj−μkmmu_(i)+mu_(j)-mu_(k_(m))\mu_{i}+\mu_{j}-\mu_{k_{m}}μi+μj−μkm monopoles have bubbled off [1]. It has a vanishing cycle FkmFkmF_(k_(m))F_{k_{m}}Fkm, corresponding to the representation VkmVkmV_(k_(m))V_{k_{m}}Vkm, as its zero section. (Viewing XXX\mathcal{X}X as the Coulomb branch, monopole bubbling is related to partial Higgsing phenomena.)
4.3.3.
Conformal blocks which diagonalize the action of braiding do not in general come from actual objects of the derived category DxDxD_(x)\mathscr{D}_{x}Dx. As is well known from Picard-Lefshetz theory, eigensheaves of braiding, branes on which the braiding acts only by degree shifts BE=E[DE]{CE}BE=EDECEBE=E[D_(E)]{C_(E)}\mathscr{B} \mathcal{E}=\mathcal{E}\left[D_{\mathcal{E}}\right]\left\{C_{\mathcal{E}}\right\}BE=E[DE]{CE}, are very rare.
by the order of vanishing of the ΠÎ Pi\PiÎ -stability central charge Z0:K(X)→CZ0:K(X)→CZ^(0):K(X)rarrC\mathcal{Z}^{0}: K(\mathcal{X}) \rightarrow \mathbb{C}Z0:K(X)→C. More precisely, one gets a pair of such filtrations, one on each side of the wall. Crossing the wall preserves the filtration, but has the effect of mixing up branes at a given order in the filtration, with those at lower orders, whose central charge vanishes faster. Because XXX\mathcal{X}X is hyper-Kahler, the ΠÎ Pi\PiÎ -stability central charge is given in terms of classical geometry (by Eq. (4.7) of [1]).
The existence of the filtration with the stated properties follows from the existence of the equivariant central charge function ZZZ\mathcal{Z}Z,
and the fact the action of braiding on KT(X)KT(X)K_(T)(X)K_{T}(\mathcal{X})KT(X) lifts to the action on DXDXDX\mathscr{D} XDX, by the theorem of [14]. The equivariant central charge ZZZ\mathcal{Z}Z is computed by the equivariant Gromov-Witten theory on XXX\mathcal{X}X in a manner analogous to VVV\mathcal{V}V, starting with the σσsigma\sigmaσ-model on the cigar DDDDD except with no insertion at its tip. It reduces to the ΠÎ Pi\PiÎ -stability central charge Z0Z0Z^(0)\mathcal{Z}^{0}Z0 by turning the equivariant parameters off.
4.3.4.
While BBB\mathscr{B}B has few eigensheaves in DxDxDx\mathscr{D} \mathcal{x}Dx, it acts by degree shifts
on the quotient subcategories. The degree shifts may be read off from the eigenvectors of the action of braiding on the equivariant central charge function ZZZ\mathcal{Z}Z. As the punctures at aiaia_(i)a_{i}ai and ajaja_(j)a_{j}aj come together, the eigenvector corresponding to the representation VkmVkmV_(k_(m))V_{k_{m}}Vkm in (4.6), vanishes as [1]
The cohomological degree shift Dkm=dimCFkmDkm=dimCâ¡FkmD_(k_(m))=dim_(C)F_(k_(m))D_{k_{m}}=\operatorname{dim}_{\mathbb{C}} F_{k_{m}}Dkm=dimCâ¡Fkm is by the dimension of the vanishing cycle. The equivariant degree shift CkmCkmC_(k_(m))C_{k_{m}}Ckm is essentially the one familiar from the action of braiding on conformal blocks of Lg^Lg^widehat(L_(g))\widehat{L_{\mathrm{g}}}Lg^ in the fusion basis [1].
4.3.5.
The derived equivalences of this type are the perverse equivalences of Chuang and Rouquier [23, 24]. They envisioned them as a way to describe derived equivalences which come from variations of Bridgeland stability conditions, but with few examples from geometry.
Traditionally, braid group actions on derived categories of coherent sheaves, or Bbranes, are fairly difficult to describe, see for example [20,21]. Braid group actions on the categories of A-branes are much easier to understand, via Picard-Lefshetz theory and its categorical uplifts [71], see e.g. [52,77]. The theory of variations of stability conditions, by Douglas and Bridgeland, was invented to bridge the two [9,29][9,29][9,29][9,29][9,29].
4.3.6.
As a by-product, we learn that conformal blocks describing collections cups or caps colored by minuscule representations, come from branes in DxDxDx\mathscr{D} xDx which have a simple geometric meaning [1][1][1][1][1].
Take X=Gr0(μ1,μ1∗,…,μd,μd∗)X=Gr0μ1,μ1∗,…,μd,μd∗X=Gr_(0)^((mu_(1),mu_(1)^(**),dots,mu_(d),mu_(d)^(**)))\mathcal{X}=\operatorname{Gr}_{0}^{\left(\mu_{1}, \mu_{1}^{*}, \ldots, \mu_{d}, \mu_{d}^{*}\right)}X=Gr0(μ1,μ1∗,…,μd,μd∗) corresponding to AAA\mathcal{A}A with n=2dn=2dn=2dn=2 dn=2d punctures, colored by pairs of complex conjugate, minuscule representations ViViV_(i)V_{i}Vi and Vi∗Vi∗V_(i)^(**)V_{i}^{*}Vi∗. We get a vanishing cycle UUUUU in XXX\mathcal{X}X which is a product of ddddd minuscule Grassmannians,
U=G/P1×⋯×G/PdU=G/P1×⋯×G/PdU=G//P_(1)xx cdots xx G//P_(d)U=G / P_{1} \times \cdots \times G / P_{d}U=G/P1×⋯×G/Pd
where PiPiP_(i)P_{i}Pi is the maximal parabolic subgroup of GGGGG associated to representation ViViV_(i)V_{i}Vi. This vanishing cycle embeds in XXX\mathcal{X}X as a compact holomorphic Lagrangian, so in the neighborhood of UUUUU, we can model XXX\mathcal{X}X as T∗UT∗UT^(**)UT^{*} UT∗U. The structure sheaf
of UUUUU is the brane we are after. The Grassmannian G/PiG/PiG//P_(i)G / P_{i}G/Pi is the cycle that vanishes when a single pair of singular monopoles of charges μiμimu_(i)\mu_{i}μi and μi∗μi∗mu_(i)^(**)\mu_{i}^{*}μi∗ come together, as Gr0(μi,μi∗)=Gr0μi,μi∗=Gr_(0)^((mu_(i),mu_(i)^(**)))=\operatorname{Gr}_{0}^{\left(\mu_{i}, \mu_{i}^{*}\right)}=Gr0(μi,μi∗)=T∗G/PiT∗G/PiT^(**)G//P_(i)T^{*} G / P_{i}T∗G/Pi.
The brane UUU\mathcal{U}U lives at the very bottom of a ddddd-fold filtration which DxDxDx\mathscr{D} xDx develops at the intersection of ddddd walls in the Kahler moduli of XXX\mathcal{X}X corresponding to bringing punctures together pairwise. It follows UUU\mathcal{U}U is the eigensheaf of braiding each pair of matched endpoints. It is extremely special, for the same reason the trivial representation is special.
4.3.7.
Just as fusion provides the right language to understand the action of braiding in conformal field theory, the perverse filtrations provide the right language to describe the action of braiding on derived categories. Using perverse filtrations and the very special properties of the vanishing cycle branes U∈DxU∈DxU inDxU \in \mathscr{D} xU∈Dx, one gets the following theorem [1]:
Theorem 1. For any simply laced Lie algebra LgLg^(L)g{ }^{L} \mathfrak{g}Lg, the homology groups
HomD∗,∗(BU,U)HomD∗,∗â¡(BU,U)Hom_(D)^(**,**)(BU,U)\operatorname{Hom}_{\mathscr{D}}^{*, *}(\mathscr{B} U, U)HomD∗,∗â¡(BU,U)
categorify Uq(Lg)UqLgU_(q)(^(L)g)U_{\mathfrak{q}}\left({ }^{L} \mathfrak{g}\right)Uq(Lg) quantum link invariants, and are themselves link invariants.
4.3.8.
As an illustration, proving that (the equivalent of) the pitchfork move in the figure below holds in DxDxDx\mathscr{D} xDx
FIGURE 3
A move equivalent to the pitchfork move.
requires showing that we have a derived equivalence
where CiCiC_(i)\mathscr{C}_{i}Ci and Ci′′Ci′′C_(i)^('')\mathscr{C}_{i}^{\prime \prime}Ci′′ are cup functors on the right and the left in Figure 3, respectively. They increase the number of strands by two and map
Ci:DXn−2→DXn and Ci′′:DXn−2→DXn′′Ci:DXn−2→DXn and Ci′′:DXn−2→DXn′′C_(i):D_(X_(n-2))rarrD_(X_(n))quad" and "quadC_(i)^(''):D_(X_(n-2))rarrD_(X_(n)^(''))\mathscr{C}_{i}: \mathscr{D}_{X_{n-2}} \rightarrow \mathscr{D}_{X_{n}} \quad \text { and } \quad \mathscr{C}_{i}^{\prime \prime}: \mathscr{D}_{X_{n-2}} \rightarrow \mathscr{D}_{X_{n}^{\prime \prime}}Ci:DXn−2→DXn and Ci′′:DXn−2→DXn′′
where the subscript serves to indicate the number of strands. The functor BBB\mathscr{B}B is the equivalence of categories from the theorem of [14]
corresponding to braiding Vk(ak)VkakV_(k)(a_(k))V_{k}\left(a_{k}\right)Vk(ak) with Vi(ai)⊗Vi∗(aj)Viai⊗Vi∗ajV_(i)(a_(i))oxV_(i)^(**)(a_(j))V_{i}\left(a_{i}\right) \otimes V_{i}^{*}\left(a_{j}\right)Vi(ai)⊗Vi∗(aj) where ViViV_(i)V_{i}Vi and Vi∗Vi∗V_(i)^(**)V_{i}^{*}Vi∗ color the red and VkVkV_(k)V_{k}Vk the black strand in Figure 3.
To prove the identity (4.10) note that
(4.11)CiDxn−2⊂Dxn and Ci′′Dxn−2⊂DXn′′(4.11)CiDxn−2⊂Dxn and Ci′′Dxn−2⊂DXn′′{:(4.11)C_(i)Dx_(n-2)subDx_(n)quad" and "quadC_(i)^('')D_(x_(n-2))subD_(X_(n)^('')):}\begin{equation*}
\mathscr{C}_{i} \mathscr{D} x_{n-2} \subset \mathscr{D} x_{n} \quad \text { and } \quad \mathscr{C}_{i}^{\prime \prime} \mathscr{D}_{x_{n-2}} \subset \mathscr{D}_{X_{n}^{\prime \prime}} \tag{4.11}
\end{equation*}(4.11)CiDxn−2⊂Dxn and Ci′′Dxn−2⊂DXn′′
are the subcategories which are the bottom-most part of the double filtrations of DxnDxnDx_(n)\mathscr{D} x_{n}Dxn and DXn′′DXn′′D_(X_(n)^(''))\mathscr{D}_{X_{n}^{\prime \prime}}DXn′′, corresponding to the intersection of walls at which the three punctures come together. By the definition of perverse filtrations, the functor BBB\mathscr{B}B acts at a bottom part of a double filtration at most by degree shifts. The degree shifts are trivial too, since if they were not, the relation we are after would not hold even in conformal field theory, and we know it does. To complete the proof, one recalls that a perverse equivalence that acts by degree shifts that are trivial is an equivalence of categories [24].
Proofs of invariance under the Reidermeister 0 and the framed Reidermeister I moves are similar. The invariance under Reidermeister II and III moves follows from the theorem of [14]. One should compare this to proofs of the same relations in [20, 21], which are more technical and less general.
4.3.9.
An elementary consequence is a geometric explanation of mirror symmetry which relates the Uq(Lg)UqLgU_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) invariants of a link KKKKK and its mirror reflection K∗K∗K^(**)K^{*}K∗.
It is a consequence of a basic property of DXDXD_(X)\mathscr{D}_{X}DX, Serre duality. Serre duality implies the isomorphism of homology groups on XXX\mathcal{X}X which is a 2d2d2d2 d2d complex-dimensional Calabi-Yau manifold,
(4.12)HomDX(BU,U[M]{J0,J→})=HomDx(BU,U[2d−M]{−d−J0,−J→})(4.12)HomDXâ¡BU,U[M]J0,J→=HomDxâ¡BU,U[2d−M]−d−J0,−J→{:(4.12)Hom_(DX)(BU,U[M]{J_(0),( vec(J))})=Hom_(Dx)(BU,U[2d-M]{-d-J_(0),-( vec(J))}):}\begin{equation*}
\operatorname{Hom}_{\mathscr{D} X}\left(\mathscr{B} U, U[M]\left\{J_{0}, \vec{J}\right\}\right)=\operatorname{Hom}_{\mathscr{D} x}\left(\mathscr{B} U, U[2 d-M]\left\{-d-J_{0},-\vec{J}\right\}\right) \tag{4.12}
\end{equation*}(4.12)HomDXâ¡(BU,U[M]{J0,J→})=HomDxâ¡(BU,U[2d−M]{−d−J0,−J→})
The equivariant degree shift comes from the fact the unique holomorphic section of the canonical bundle has weight ddddd under the Cq×⊂TCq×⊂TC_(q)^(xx)subT\mathbb{C}_{\mathrm{q}}^{\times} \subset \mathrm{T}Cq×⊂T action. Mirror symmetry follows by taking Euler characteristic of both sides [1].
4.4. Algebra from B-branes
Bezrukavnikov and Kaledin, using quantization in characteristic ppppp, constructed a tilting vector bundle TTT\mathcal{T}T, on any XXX\mathcal{X}X which is a symplectic resolution [12,13,43,44][12,13,43,44][12,13,43,44][12,13,43,44][12,13,43,44]. Its endomorphism algebra
is an ordinary associative algebra, graded only by equivariant degrees. The derived category DADAD_(A)\mathscr{D}_{\mathscr{A}}DA of its modules is equivalent to DXDXD_(X)\mathscr{D}_{X}DX,
Webster recently computed the algebra AAA\mathscr{A}A for our XXX\mathcal{X}X [80], and showed that it coincides with a cylindrical version of the KLRW algebra from [78]. Working with the cylindrical KLRW algebra, as opposed to the ordinary one, leads to invariants of links in R2×S1R2×S1R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1 and not just in R3R3R^(3)\mathbb{R}^{3}R3. The KLRW algebra generalizes the algebras of Khovanov and Lauda [50] and Rouquier [68]. The cylindrical version of the KLR algebra corresponds to XXX\mathcal{X}X which is a Coulomb branch of a pure 3D gauge theory.
4.4.1.
The description of link homologies via DX=CohT(X)DX=CohTâ¡(X)D_(X)=Coh_(T)(X)\mathscr{D}_{X}=\operatorname{Coh}_{T}(\mathcal{X})DX=CohTâ¡(X) provides a geometric meaning of homological Uq(Lg)UqLgU_(q)(^(L)g)U_{\mathfrak{q}}\left({ }^{L} \mathfrak{g}\right)Uq(Lg) link invariants. Even so, without further input, the description of link homologies either in terms of DXDXD_(X)\mathscr{D}_{X}DX or DADAD_(A)\mathscr{D}_{\mathscr{A}}DA is purely formal. With the help of (equivariant) homological mirror symmetry, we will give a description of link homology groups which is explicit and explicitly computable; in this sense, link homology groups come to life in the mirror.
5. MIRROR SYMMETRY FOR MONOPOLE MODULI SPACE
In the very best instances, homological mirror symmetry relating DyDyDy\mathscr{D} yDy and DxDxDx\mathscr{D} xDx can be made manifest, by showing that each is equivalent to DADAD_(A)\mathscr{D}_{\mathscr{A}}DA, the derived category of modules
of the same associative algebra AAA\mathscr{A}A,
(5.1)Dx≅DA≅Dy(5.1)Dx≅DA≅Dy{:(5.1)D_(x)~=D_(A)~=Dy:}\begin{equation*}
\mathscr{D}_{x} \cong \mathscr{D}_{\mathscr{A}} \cong \mathscr{D} y \tag{5.1}
\end{equation*}(5.1)Dx≅DA≅Dy
is the endomorphism algebra of a set of branes T=⨁CTeT=â¨C TeT=bigoplus_(C)Te\mathcal{T}=\bigoplus_{\mathscr{C}} \mathcal{T} \mathscr{e}T=â¨CTe, which generate DxDxDx\mathscr{D} xDx and DyDyDy\mathscr{D} yDy. For economy, we will be denoting branes related by mirror symmetry by the same letter.
An elementary example [10] is mirror symmetry relating a pair of infinite cylinders, X=C×X=C×X=C^(xx)\mathcal{X}=\mathbb{C}^{\times}X=C×and y=R×S1y=R×S1y=RxxS^(1)y=\mathbb{R} \times S^{1}y=R×S1, whose torus fibers are dual S1S1S^(1)S^{1}S1, s. Both DxDxDx\mathscr{D} xDx, the derived category of coherent sheaves on XXX\mathcal{X}X, and DyDyDy\mathscr{D} yDy, based on the wrapped Fukaya category, are generated by a single object TTT\mathcal{T}T, a flat line bundle on XXX\mathcal{X}X and a real-line Lagrangian on YYY\mathscr{Y}Y. Their algebras of open strings are the same, equal to the algebra A=C[x±1]A=Cx±1A=C[x^(+-1)]\mathscr{A}=\mathbb{C}\left[x^{ \pm 1}\right]A=C[x±1] of holomorphic functions on the cylinder.
XXX\mathcal{X}X
FIGURE 4
A simple example of manifest mirror symmetry.
5.1. The algebra for homological mirror symmetry
In our setting, the generator TTT\mathcal{T}T of DxDxDx\mathscr{D} xDx is the tilting generator of Bezrukavnikov and Kaledin from Section 4.4. Webster's proof of the equivalence of categorification of Uq(Lg)UqLgU_(q)(^(L)g)U_{\mathfrak{q}}\left({ }^{L} \mathfrak{g}\right)Uq(Lg) link invariants and B-type branes on XXX\mathcal{X}X and via the cKLRW algebra AAA\mathscr{A}A should be understood as the first of the two equivalences in (5.1).
5.1.1.
The mirror yyyyy of XXX\mathcal{X}X is the moduli space of GGGGG monopoles, of the same charges as XXXXX except on R2×S1R2×S1R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1 instead of on R3R3R^(3)\mathbb{R}^{3}R3, with only complex and no Kahler moduli turned on, and equipped with a potential [2]. Without the potential, the mirror to yyyyy would be another moduli space of GGGGG monopoles on R2×S1R2×S1R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1.
The theory based on DyDyDy\mathscr{D} yDy, the derived Fukaya-Seidel category of yyyyy, is in the same spirit as the work of Seidel and Smith [72]. They pioneered geometric approaches to link homology, but produced a only singly graded theory, known as symplectic Khovanov homology. The computation of DyDyDy\mathscr{D} yDy, which makes mirror symmetry in (5.1) manifest, is given in the joint work with Danilenko, Li, and Zhou [4].
5.2. The core of the monopole moduli space
Working equivariantly with respect to a Cq×Cq×C_(q)^(xx)\mathbb{C}_{\mathrm{q}}^{\times}Cq×-symmetry which scales the holomorphic symplectic form of XXX\mathcal{X}X, all the information about its geometry should be encoded in a core locus preserved by such actions.
The core XXXXX is a singular holomorphic Lagrangian in XXX\mathcal{X}X which is the union of supports of all stable envelopes [7,61]. Equivalently, XXXXX is the union of all attracting sets of ΛΛLambda\LambdaΛ-torus actions on XXX\mathcal{X}X, where we let ΛΛLambda\LambdaΛ vary over all chambers. If we view XXX\mathcal{X}X as the monopole moduli space, we can put this more simply: XXXXX is the locus where all the monopoles, singular or not, are at the origin of CCC\mathbb{C}C in R×CR×CRxxC\mathbb{R} \times \mathbb{C}R×C. Viewing it as a Coulomb branch, XXXXX is the locus at which the complex scalar fields in vector multiplets vanish.
We will define the equivariant mirror YYYYY of XXX\mathcal{X}X to be the ordinary mirror of its core, so we have
Working equivariantly with respect to the TTTTT-action on XXX\mathcal{X}X, the equivariant mirror gets a potential WWWWW, making the theory on YYYYY into a Landau-Ginsburg model. While XXXXX embeds into XXX\mathcal{X}X as a holomorphic Lagrangian of dimension d,yd,yd,yd, yd,y fibers over YYYYY with holomorphic Lagrangian (C×)dC×d(C^(xx))^(d)\left(\mathbb{C}^{\times}\right)^{d}(C×)d fibers.
5.2.1.
A model example is XXX\mathcal{X}X which is the resolution of an An−1An−1A_(n-1)A_{n-1}An−1 hypersurface singularity, uv=zn;Xuv=zn;Xuv=z^(n);Xu v=z^{n} ; \mathcal{X}uv=zn;X is the moduli space of a single smooth G=SU(2)/Z2G=SU(2)/Z2G=SU(2)//Z_(2)G=\mathrm{SU}(2) / \mathbb{Z}_{2}G=SU(2)/Z2 monopole, in the presence of nnnnn singular ones. The core XXXXX is a collection of n−1P1n−1P1n-1P^(1)n-1 \mathbb{P}^{1}n−1P1 's with a pair of infinite discs attached, as in Figure 5.
FIGURE 5
Core XXXXX of a resolution of the An−1An−1A_(n-1)A_{n-1}An−1 singularity.
The ordinary mirror yyyyy of XXX\mathcal{X}X is the complex structure deformation of the "multiplicative" An−1An−1A_(n-1)A_{n-1}An−1 surface singularity, with a potential which we will not need. yyyyy is a C×C×C^(xx)\mathbb{C}^{\times}C×fibration over YYYYY which is itself an infinite cylinder, a copy of C×C×C^(xx)\mathbb{C}^{\times}C×with nnnnn points deleted. At the marked points, the C×C×C^(xx)\mathbb{C}^{\times}C×fibers degenerate. There are n−1n−1n-1n-1n−1 Lagrangian spheres in yyyyy, which are mirror to n−1P1n−1P1n-1P^(1)n-1 \mathbb{P}^{1}n−1P1 's in XXXXX. They project to Lagrangians in YYYYY which begin and end at the punctures.
5.2.2.
The model example corresponds to LG=SU(2)LG=SU(2)^(L)G=SU(2){ }^{L} G=\mathrm{SU}(2)LG=SU(2) Chern-Simons theory on R2×S1R2×S1R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1, and N12^N12^widehat(N1_(2))\widehat{\mathfrak{N 1}_{2}}N12^ conformal blocks on A=R×S1A=R×S1A=RxxS^(1)\mathcal{A}=\mathbb{R} \times S^{1}A=R×S1. The nnnnn punctures on AAA\mathcal{A}A are colored by the fundamental, two-dimensional representation V1/2V1/2V_(1//2)V_{1 / 2}V1/2 of Nu2Nu2Nu_(2)\mathfrak{N u}_{2}Nu2, and we take the subspace of weight
FIGURE 6
Lagrangian spheres in yyy\boldsymbol{y}y mirror the vanishing P1P1P^(1)\mathbb{P}^{1}P1 's in XXX\mathcal{X}X.
one level below the highest. Note that YYYYY coincides with the Riemann surface AAA\mathcal{A}A where the conformal blocks live. This is not an accident.
In the model example, both XXXXX and YYYYY are S1S1S^(1)S^{1}S1 fibrations over RRR\mathbb{R}R with nnnnn marked points. At the marked points, the S1S1S^(1)S^{1}S1 fibers of XXXXX degenerate. In YYYYY, this is mirrored by fibers that decompactify, due to points which are deleted.
5.2.3.
More generally, for X=Grμ→νX=Grμ→νX=Gr^( vec(mu))_(nu)\mathcal{X}=\mathrm{Gr}^{\vec{\mu}}{ }_{\nu}X=Grμ→ν we have dadad_(a)d_{a}da smooth GGGGG-monopoles colored by simple roots LeaLea^(L)e_(a){ }^{L} e_{a}Lea and otherwise identical. It follows that the common base of SYZ fibrations of XXXXX and YYYYY is the configuration space of the smooth monopoles on the real line RRR\mathbb{R}R with nnnnn marked points. The marked points are labeled by the weights μiμimu_(i)\mu_{i}μi of LgLg^(L)g{ }^{L} \mathrm{~g}L g, which are the singular monopole charges.
An explicit description of YYYYY, as well as its category of A-branes DYDYD_(Y)\mathscr{D}_{Y}DY, is given [4]. Here we will only describe some of its features. In an open set, YYYYY coincides with
the configuration space of d=∑a=1rkdad=∑a=1rk dad=sum_(a=1)^(rk)d_(a)d=\sum_{a=1}^{r k} d_{a}d=∑a=1rkda points on the punctured Riemann surface AAA\mathcal{A}A, "colored" by simple roots LeaLea^(L)e_(a){ }^{L} e_{a}Lea of LgLg^(L)g{ }^{L} \mathrm{~g}L g, but otherwise identical. The open set is the complement of the divisor of zeros and of poles of function f0f0f^(0)f^{0}f0 in (5.5).
This allows DYDYD_(Y)\mathscr{D}_{Y}DY to have a ZZZ\mathbb{Z}Z-valued cohomological grading. The symplectic form on YYYYY is inherited from the symplectic form on yyyyy, by restricting it to the vanishing (S1)dS1d(S^(1))^(d)\left(S^{1}\right)^{d}(S1)d in each of its (C×)dC×d(C^(xx))^(d)\left(\mathbb{C}^{\times}\right)^{d}(C×)d fibers over YYYYY [4]. The precise choice of symplectic structure is the one compatible
with mirror symmetry which we used to define YYYYY, as the equivariant mirror of X=Grμ→νX=Grμ→νX=Gr^( vec(mu))_(nu)\mathcal{X}=\mathrm{Gr}^{\vec{\mu}}{ }_{\nu}X=Grμ→ν and the ordinary mirror of its core.
Including the equivariant TTTTT-equivariant action on XXX\mathcal{X}X and XXXXX corresponds to adding to the σσsigma\sigmaσ-model on YYYYY a potential
which is a multivalued holomorphic function on Y;λaY;λaY;lambda_(a)Y ; \lambda_{a}Y;λa are the equivariant parameters of the ΛΛLambda\LambdaΛ-action on XXX\mathcal{X}X, and
q=e2πiλ0q=e2Ï€iλ0q=e^(2pi ilambda_(0))q=e^{2 \pi i \lambda_{0}}q=e2Ï€iλ0
The potentials W0W0W^(0)W^{0}W0 and WaWaW^(a)W^{a}Wa are given by
The superpotential WWWWW breaks the conformal invariance of the σσsigma\sigmaσ-model to YYYYY if λ0≠0λ0≠0lambda_(0)!=0\lambda_{0} \neq 0λ0≠0, since only a quasihomogenous superpotential is compatible with it. This is mirror to breaking of conformal invariance on XXX\mathcal{X}X by the Cq×Cq×C_(q)^(xx)\mathbb{C}_{q}^{\times}Cq×-action for q≠1q≠1q!=1q \neq 1q≠1.
Since W0W0W^(0)W^{0}W0 and WaWaW^(a)W^{a}Wa are multivalued, YYYYY is equipped with a collection of closed oneforms with integer periods
c0=dW0/2πi,ca=dWa/2πi∈H1(Y,Z)c0=dW0/2Ï€i,ca=dWa/2Ï€i∈H1(Y,Z)c^(0)=dW^(0)//2pi i,quadc^(a)=dW^(a)//2pi i inH^(1)(Y,Z)c^{0}=d W^{0} / 2 \pi i, \quad c^{a}=d W^{a} / 2 \pi i \in H^{1}(Y, \mathbb{Z})c0=dW0/2Ï€i,ca=dWa/2Ï€i∈H1(Y,Z)
which introduce additional gradings in the category of A-branes, as in [73].
5.2.4.
From the mirror perspective, the conformal blocks of Lg^Lg^widehat(L_(g))\widehat{L_{\mathrm{g}}}Lg^ come from the B-twisted Landau-Ginsburg model (Y,W)(Y,W)(Y,W)(Y, W)(Y,W) on DDDDD which is an infinitely long cigar, with A-type boundary condition at infinity corresponding to a Lagrangian L∈YL∈YL in YL \in YL∈Y. The partition function of the theory has the following form:
We have (re)discovered, from mirror symmetry, an integral representation of the conformal blocks of Lg^Lg^widehat(L_(g))\widehat{L_{\mathrm{g}}}Lg^. This "free field representation" of conformal blocks, remarkable for its simplicity [32], goes back to the 1980s work of Kohno and Feigin and Frenkel [34,54], and of Schechtman and Varchenko [69,70].
5.2.5.
There is a reconstruction theory, due to Givental [38] and Teleman [76], which says that, starting with the solution of the quantum differential equation or its mirror counterpart, one gets to reconstruct all genus topological string amplitudes for any semisimple 2D field theory. The semisimplicity condition is satisfied in our case, as WWWWW has isolated critical points. It follows the B-twisted Landau-Ginsburg model on (Y,W)(Y,W)(Y,W)(Y, W)(Y,W) and A-twisted TTTTT-equivariant sigma model on XXX\mathcal{X}X are equivalent to all genus [2]. Thus, equivariant mirror symmetry holds as an equivalence of topological string amplitudes.
5.3. Equivariant Fukaya-Seidel category
For every A-brane LLLLL at the boundary at infinity of the cigar DDDDD, we get a solution of the KZKZKZ\mathrm{KZ}KZ equation. The brane is an object of
the derived Fukaya-Seidel category of YYYYY with potential WWWWW. The category should be thought of as a category of equivariant A-branes, due to the fact WWWWW in (5.4) is multivalued. Another novel aspect of DYDYD_(Y)\mathscr{D}_{Y}DY is that it provides an example of Fukaya-Seidel category with coefficients in perverse schobers. This structure, inherited from equivariant mirror symmetry, was discovered in [4][4][4][4][4].
5.3.1.
Objects of DYDYD_(Y)\mathscr{D}_{Y}DY are Lagrangians in YYYYY, equipped with some extra data. A Lagrangian in YYYYY is a product of ddddd one-dimensional curves on AAA\mathscr{A}A which are colored by simple roots and may be immersed; or a simplex obtained from an embedded curve, as a configuration space of ddddd partially ordered colored points. The theory also includes more abstract branes, which are iterated mapping cones over morphisms between Lagrangians.
5.3.2.
The extra data includes a grading by Maslov and equivariant degrees. The equivariant grading of a brane in DYDYD_(Y)\mathscr{D}_{Y}DY is defined by choosing a lift of the phase of e−We−We^(-W)e^{-W}e−W to a real-valued function on the Lagrangian LLLLL. The equivariant degree shift operation,
More generally, branes in DYDYD_(Y)\mathscr{D}_{Y}DY are graded Lagrangians LLLLL equipped with an extra structure of a local system ΛΛLambda\LambdaΛ of modules of a certain algebra BBB\mathscr{B}B we will describe shortly. For the time being, only branes for which ΛΛLambda\LambdaΛ is trivial will play a role for us.
5.3.3.
The space of morphisms between a pair of Lagrangian branes in a derived Fukaya category
is defined by Floer theory, which itself is modeled after Morse theory approach to supersymmetric quantum mechanics, from the introduction. The role of the Morse complex is taken by the Floer complex.
For branes equipped with a trivial local system, the Floer complex
is a graded vector space spanned by the intersection points of the two Lagrangians, together with the action of a differential QQQQQ. The complex is graded by the fermion number, which is the Maslov index, and the equivariant gradings, thanks to the fact WWWWW is multivalued.
is generated by instantons. In Floer theory, the coefficient of P′P′P^(')\mathcal{P}^{\prime}P′ in QPQPQPQ \mathcal{P}QP is obtained by "counting" holomorphic strips in YYYYY with boundary on L0L0L_(0)L_{0}L0 and L1L1L_(1)L_{1}L1, interpolating from PPP\mathscr{P}P to P′P′P^(')\mathcal{P}^{\prime}P′, of Maslov index 1 and equivariant degree 0 . The cohomology of the resulting complex is Floer cohomology.
5.3.4.
A simplification in the present case is that, just as branes have a description in terms of the Riemann surface, so do their intersection points, as well as the maps between them.
The theory that results is a generalization of Heegard-Floer theory, which is associated to Lg=gl1∣1Lg=gl1∣1^(L)g=gl_(1∣1){ }^{L} \mathfrak{g}=\mathfrak{g l}_{1 \mid 1}Lg=gl1∣1 and categorifies the Alexander polynomial [63,64]. Heegard-Floer theory
thought of as a configuration space of fermions on the Riemann surface, as opposed to anyons for YSu2YSu2Y_(Su_(2))Y_{\mathfrak{S u}_{2}}YSu2; in particular, their top holomorphic forms differ.
While we so far assumed that LgLg^(L)g{ }^{L} \mathrm{~g}L g is simply laced, the DYDYD_(Y)\mathscr{D}_{Y}DY has an extension to nonsimply-laced Lie algebras, as well as glm∣nglm∣ngl_(m∣n)\mathrm{gl}_{m \mid n}glm∣n and sppm∣2nsppm∣2nspp_(m∣2n)\mathfrak{s p} \mathfrak{p}_{m \mid 2 n}sppm∣2n Lie superalgebras, described in [3,5].
5.4. Link invariants and equivariant mirror symmetry
Mirror symmetry helps us understand exactly which questions we need to ask to recover homological knot invariants from YYYYY.
5.4.1.
Since YYYYY is the ordinary mirror of XXXXX, we should start by understanding how to recover homological knot invariants from XXXXX, rather than XXX\mathcal{X}X. Every B-brane on XXX\mathcal{X}X which is relevant for us comes from a BBBBB-brane on XXXXX via an exact functor
(5.8)f∗:DX→Dx(5.8)f∗:DX→Dx{:(5.8)f_(**):D_(X)rarrDx:}\begin{equation*}
f_{*}: \mathscr{D}_{X} \rightarrow \mathscr{D} x \tag{5.8}
\end{equation*}(5.8)f∗:DX→Dx
which interprets a sheaf "downstairs" on XXXXX as a sheaf "upstairs" on XXX\mathcal{X}X. The functor f∗f∗f_(**)f_{*}f∗ is more precisely the right-derived functor Rf∗Rf∗Rf_(**)R f_{*}Rf∗. Its adjoint
is the left derived functor Lf∗Lf∗Lf^(**)L f^{*}Lf∗, and corresponds to tensoring with the structure sheaf ⊗OX⊗OXoxO_(X)\otimes \mathcal{O}_{X}⊗OX, and restricting. Adjointness implies that, given any pair of branes on XXX\mathcal{X}X that come from XXXXX,
after replacing FFFFF with f∗f∗Ff∗f∗Ff^(**)f_(**)Ff^{*} f_{*} Ff∗f∗F. The functor f∗f∗f∗f∗f^(**)f_(**)f^{*} f_{*}f∗f∗ is not identity on DXDXD_(X)\mathscr{D}_{X}DX.
5.4.2.
The equivariant homological mirror symmetry relating DXDXD_(X)\mathscr{D}_{X}DX and DYDYD_(Y)\mathscr{D}_{Y}DY is not an equivalence of categories, but a correspondence of branes and Hom's which come from a pair of adjoint functors h∗h∗h_(**)h_{*}h∗ and h∗h∗h^(**)h^{*}h∗, inherited from f∗f∗f_(**)f_{*}f∗ and f∗f∗f^(**)f^{*}f∗ via the downstairs homological mirror symmetry:
Alternatively, h∗h∗h^(**)h^{*}h∗ and h∗h∗h_(**)h_{*}h∗ come by composing the upstairs mirror symmetry with a pair of functors k∗:Dy→DYk∗:Dy→DYk^(**):Dy rarrD_(Y)k^{*}: \mathscr{D} y \rightarrow \mathscr{D}_{Y}k∗:Dy→DY and k∗:DY→Dyk∗:DY→Dyk_(**):D_(Y)rarrDyk_{*}: \mathscr{D}_{Y} \rightarrow \mathscr{D} yk∗:DY→Dy, which are mirror to f∗f∗f^(**)f^{*}f∗ and f∗f∗f_(**)f_{*}f∗. The functors k∗,k∗k∗,k∗k^(**),k_(**)k^{*}, k_{*}k∗,k∗ come from Lagrangian correspondences; their construction is described in joint work with McBreen, Shende, and Zhou [6]. The functor k∗k∗k_(**)k_{*}k∗ amounts to pairing a brane downstairs, with a vanishing torus fiber over it; this is how Figure 6 arises in our model example. The adjoint functors let us recover answers to all interesting questions about XXX\mathcal{X}X from YYYYY.
5.4.3.
For any simply laced Lie algebra LgLg^(L)g{ }^{L} \mathrm{~g}L g, the branes U∈DXU∈DXU inDXU \in \mathscr{D} XU∈DX which serve as cups upstairs are the structure sheaves of (products of) minuscule Grassmannians, as described in Section 4.3.6. They come via the functor h∗h∗h_(**)h_{*}h∗ from branes IU∈DYIU∈DYIU inD_(Y)I U \in \mathscr{D}_{Y}IU∈DY downstairs, on YYYYY
U=h∗IUU=h∗IUU=h_(**)IUU=h_{*} I UU=h∗IU
which are (products of) generalized intervals. A minuscule Grassmannian G/PiG/PiG//P_(i)G / P_{i}G/Pi is the h∗−h∗−h_(**^(-))h_{*^{-}}h∗− image of a brane which is the configuration space of colored points on an interval ending on a pair of punctures on AAA\mathscr{A}A corresponding to representations ViViV_(i)V_{i}Vi and Vi∗Vi∗V_(i)^(**)V_{i}^{*}Vi∗. The points are colored by simple positive roots in μi+μi∗=∑ada,iLeaμi+μi∗=∑a da,iLeamu_(i)+mu_(i)^(**)=sum_(a)d_(a,i)^(L)e_(a)\mu_{i}+\mu_{i}^{*}=\sum_{a} d_{a, i}{ }^{L} e_{a}μi+μi∗=∑ada,iLea, and ordered in the sequence by which, to obtain the lowest weight μi∗μi∗mu_(i)^(**)\mu_{i}^{*}μi∗ in representation ViViV_(i)V_{i}Vi, we subtract simple positive
FIGURE 7
The cup and cap A-branes corresponding to the defining representation of Lg=su4Lg=su4^(L)g=su_(4){ }^{L} \mathfrak{g}=\mathfrak{s u}_{4}Lg=su4, colored by its three simple roots; they are equivariant mirror to a B-brane supported on a P4P4P^(4)\mathbb{P}^{4}P4 as its structure sheaf.
roots from the highest weight μiμimu_(i)\mu_{i}μi. Because ViViV_(i)V_{i}Vi and Vi∗Vi∗V_(i)^(**)V_{i}^{*}Vi∗ are minuscule representations, the ordering and hence the brane IUIUIUI UIU is unique, up to equivalence and a choice of grading. The UUU\mathcal{U}U branes project back down as generalized figure-eight branes; these are nested products of figure-eights, colored by simple roots
h∗U=h∗h∗IU=EUh∗U=h∗h∗IU=EUh^(**)U=h^(**)h_(**)IU=EUh^{*} U=h^{*} h_{*} I U=E Uh∗U=h∗h∗IU=EU
and ordered analogously, as in Figure 7. As objects of DYDYD_(Y)\mathscr{D}_{Y}DY, these branes are best described iterated cones over more elementary branes, mirror to stable basis branes [5]. The cup and cap branes all come with trivial local systems, for which the Floer complexes are the familiar ones, given by (5.7).
As an example, for Lg=SH2Lg=SH2^(L)g=SH_(2){ }^{L} \mathfrak{g}=\mathfrak{S H}_{2}Lg=SH2 the only minuscule representation is the defining representation Vi=V12Vi=V12V_(i)=V_((1)/(2))V_{i}=V_{\frac{1}{2}}Vi=V12, which is self-conjugate. The cup brane UUUUU in XXX\mathcal{X}X is a product of ddddd non-intersecting P1P1P^(1)\mathbb{P}^{1}P1, s. It comes, as the image of h∗h∗h_(**)h_{*}h∗, from a brane IUIUIUI UIU in YYYYY which is a product of ddddd simple intervals, connecting pairs of punctures that come together. The UUU\mathcal{U}U-brane projects back down, via the h∗h∗h^(**)h^{*}h∗ functor, as a product of ddddd elementary figure-eight branes. The branes are graded by Maslov and equivariant gradings, as described in [2].
5.4.4.
In the description based on YYYYY, both the branes, and the action of braiding on them is geometric, so we can simply start with a link and a choice of projection to the surface A=R×S1A=R×S1A=RxxS^(1)\mathcal{A}=\mathbb{R} \times S^{1}A=R×S1. A link contained in a three ball in R2×S1R2×S1R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1 is equivalent to the same link in R3R3R^(3)\mathbb{R}^{3}R3, and projects to a contractible patch on AAA\mathcal{A}A.
To translate the link to a pair of A-branes, start by choosing bicoloring of the link projection, such that each of its components has an equal number of red and blue segments, and the red always underpass the blue. For a link component colored by a representation ViViV_(i)V_{i}Vi of LgLg^(L)g{ }^{L} \mathrm{~g}L g, place a puncture colored by its highest weight μiμimu_(i)\mu_{i}μi where a blue segment begins and its conjugate μi∗μi∗mu_(i)^(**)\mu_{i}^{*}μi∗ where it ends; the orientation of the link component distinguishes the two. The mirror Lagrangians IuIuI_(u)I_{u}Iu and BEUBEUBE_(U)\mathscr{B} E_{\mathcal{U}}BEU are obtained by replacing all the blue segments by the interval branes, and all the red segments by figure-eight branes, related by equivariant mirror symmetry to minuscule Grassmannian branes. This data determines both YYYYY and the branes on it we need. The variant of the second step, applicable for Lie superalgebras, is described in [5].
FIGURE 8
A bicoloring of the left-handed trefoil.
Equivariant mirror symmetry predicts that a homological link invariant is the space of morphisms
the cohomology of the Floer complex of the two branes. In what follows, will explain how to compute it.
FIGURE 9
The branes corresponding to the left-handed trefoil in Lg=su2Lg=su2^(L)g=su_(2){ }^{L} \mathfrak{g}=\mathfrak{s u}_{2}Lg=su2. The knot was isotoped relative to Figure 8 .
5.4.5.
To evaluate the Euler characteristic of the homology in (5.11), one simply counts intersection points of Lagrangians, keeping track of gradings. For links in R3R3R^(3)\mathbb{R}^{3}R3, the equivariant grading in (5.11) collapses to a ZZZ\mathbb{Z}Z-grading. The Euler characteristic becomes
where M(P)M(P)M(P)M(\mathscr{P})M(P) and J(P)J(P)J(P)J(\mathscr{P})J(P) are the Maslov and c0c0c^(0)c^{0}c0-grading of the point PPP\mathscr{P}P; as in Heegard-Floer theory, there are purely combinatorial formulas for them [3,5]. Mirror symmetry implies that this is the Uq(Lg)UqLgU_(q)(^(L)(g))U_{\mathfrak{q}}\left({ }^{L} \mathrm{~g}\right)Uq(LÂ g) invariant of the link in R3R3R^(3)\mathbb{R}^{3}R3.
The fact that for Lg=su2Lg=su2^(L)g=su_(2){ }^{L} \mathfrak{g}=\mathfrak{s u}_{2}Lg=su2 the graded count of intersection points in (5.12) reproduces the Jones polynomial is a theorem of Bigelow [15], building on the work of Lawrence [56-58]. Bigelow also proved the statement for Lg=suNLg=suN^(L)g=su_(N){ }^{L} \mathfrak{g}=\mathfrak{s u}_{N}Lg=suN with links colored by the defining representation [16]. The equivariant homological mirror symmetry explains the origin of Bigelow's peculiar construction, and generalizes it to other Uq(Lg)UqLgU_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(LÂ g) link invariants. 11^(1){ }^{1}1
5.4.6.
The action of the differential QQQQQ on the Floer complex, defined by counting holomorphic maps from a disk DDDDD to YYYYY with boundaries on the pair of Lagrangians, should have a reformulation [2] in terms of counting holomorphic curves embedded in D×AD×AD xxAD \times \mathcal{A}D×A with certain properties, generalizing the cylindrical formulation of Heegard-Floer theory due to Lipshitz [59]. The curve must have a projection to DDDDD as a d=∑adad=∑a dad=sum_(a)d_(a)d=\sum_{a} d_{a}d=∑ada-fold cover, with branching only between components of one color, and a projection to AAA\mathscr{A}A as a domain with boundaries on one-dimensional Lagrangians of matching colors. In addition, the potential WWWWW must pull back to DDDDD as a regular holomorphic function. Computing the action of QQQQQ in this framework reduces to solving a sequence of well defined, but hard, problems in complex analysis in one dimension, which are applications of the Riemannian mapping theorem, similar to that in [63].
proven in [4]. A basic virtue of mirror symmetry is that it sums up holomorphic curve counts. In our case, it solves all the disk-counting problems required to find the action of the differential QQQQQ on the Floer complex underlying (6.1).
6.1. The algebra of A-branes
As in the simplest examples of homological mirror symmetry, DXDXD_(X)\mathscr{D}_{X}DX and DYDYD_(Y)\mathscr{D}_{Y}DY are both generated by a finite set of branes.
6.1.1.
From perspective of YYYYY, the generating set of branes come from products of real line Lagrangians on AAA\mathscr{A}A, colored by d=∑adad=∑a dad=sum_(a)d_(a)d=\sum_{a} d_{a}d=∑ada simple roots. The brane is unchanged by reorder ing a pair of its neighboring Lagrangian components, provided they are colored by roots
is the generator of DYDYD_(Y)\mathscr{D}_{Y}DY which is mirror to the tilting vector bundle on XXXXX, which generates DXDXD_(X)\mathscr{D}_{X}DX. This generalizes the simplest example of mirror symmetry from Section 5.1. As before, we will be denoting branes on XXXXX and on YYYYY related by homological mirror symmetry by the same letter.
6.1.2.
A well known phenomenon in mirror symmetry is that it may introduce Lagrangians with an extra structure of a local system, a nontrivial flat U(1)U(1)U(1)U(1)U(1) bundle. The mirror of a structure sheaf of a generic point, in our model example of mirror symmetry from Section 5.1, is a Lagrangian of this sort.
Here, we find a generalization of this [4]. The pair of adjoint functors h∗h∗h_(**)h_{*}h∗ and h∗h∗h^(**)h^{*}h∗ that relate DYDYD_(Y)\mathscr{D}_{Y}DY with its equivariant mirror DXDXD_(X)\mathscr{D}_{X}DX equip each TTTTT-brane with a vector bundle or, more precisely, with a local system of modules for a graded algebra BBB\mathscr{B}B. The algebra is a product B=⨂a=1rkBdaB=⨂a=1rk BdaB=⨂_(a=1)^(rk)B_(d_(a))\mathscr{B}=\bigotimes_{a=1}^{r k} \mathscr{B}_{d_{a}}B=⨂a=1rkBda, where BdBdB_(d)\mathscr{B}_{d}Bd has a representation as the quotient of the algebra of polynomials in ddddd variables z1,…,zdz1,…,zdz_(1),dots,z_(d)z_{1}, \ldots, z_{d}z1,…,zd which sets their symmetric functions to zero. The zzzzz 's have equivariant qqqqq-degree equal to one.
are defined through a perturbation of TCTCT_(C)T_{\mathcal{C}}TC which induces wrapping near infinities of AAA\mathcal{A}A, as in Figure 4, and other examples of wrapped Fukaya categories.
The Floer cohomology groups HF are cohomology groups of the Floer complex whose generators are intersection points of the TeTeT_(e)T_{e}Te branes, with coefficients in BBB\mathscr{B}B. The generators all have homological degree zero, so the Floer differential is trivial, and
(6.3)HomDY(Te,Te′[k]{d→})=0, for all k≠0 and all d→(6.3)HomDYâ¡Te,Te′[k]{d→}=0, for all k≠0 and all d→{:(6.3)Hom_(D_(Y))(T_(e),T_(e^('))[k]{( vec(d))})=0","quad" for all "k!=0" and all " vec(d):}\begin{equation*}
\operatorname{Hom}_{\mathscr{D}_{Y}}\left(T_{e}, T_{\mathcal{e}^{\prime}}[k]\{\vec{d}\}\right)=0, \quad \text { for all } k \neq 0 \text { and all } \vec{d} \tag{6.3}
\end{equation*}(6.3)HomDYâ¡(Te,Te′[k]{d→})=0, for all k≠0 and all d→
The Floer product on DYDYD_(Y)\mathscr{D}_{Y}DY makes
into an algebra, which is an ordinary associative algebra, graded only by equivariant degrees.
6.1.4.
The vanishing in (6.3) mirrors the tilting property of TTTTT viewed as the generator of DXDXD_(X)\mathscr{D}_{X}DX. The tilting vector bundle T∈DXT∈DXT inD_(X)T \in \mathscr{D}_{X}T∈DX is inherited from the Bezrukavnikov-Kaledin tilting bundle TTT\mathcal{T}T on XXX\mathcal{X}X,
from Section 4.4, as the image of the f∗f∗f^(**)f^{*}f∗ functor, which is tensoring with the structure sheaf of XXXXX and restriction, f∗T=T∈DXf∗T=T∈DXf^(**)T=T inD_(X)f^{*} \mathcal{T}=T \in \mathscr{D}_{X}f∗T=T∈DX. The endomorphism of the upstairs tilting generator T ,
Since TTT\mathcal{T}T is a vector bundle on XXX\mathcal{X}X, the center of AAA\mathscr{A}A is the algebra of holomorphic functions on XXX\mathcal{X}X. The downstairs algebra is a quotient of the upstairs one, by a two-sided ideal
The ideal III\mathscr{I}I is generated by holomorphic functions that vanish on the core XXXXX.
6.1.5.
The cKLRW algebra AAA\mathscr{A}A is defined as an algebra of colored strands on a cylinder, decorated with dots, where composition is represented by stacking cylinders and rescaling [80]. The local algebra relations are those of the ordinary KLRW algebra from [78]. Placing the theory on the cylinder, it gets additional gradings by the winding number of strands of a given color, corresponding to the equivariant ΛΛLambda\LambdaΛ-action on XXX\mathcal{X}X.
The elements of the algebra A=A/IA=A/IA=A//IA=\mathscr{A} / \mathscr{I}A=A/I have a geometric interpretation by recalling the Floer complex CF∗(TC,T⨀′)CF∗TC,T⨀′ CF^(**)(T_(C),T_(⨀'))\mathrm{CF}^{*}\left(T_{\mathcal{C}}, T_{\bigodot^{\prime}}\right)CF∗(TC,T⨀′) is fundamentally generated by paths rather the intersection points. The S1S1S^(1)S^{1}S1 of the algebra cylinder is the S1S1S^(1)S^{1}S1 in the Riemann surface AAA\mathcal{A}A; its vertical direction parameterizes the path. The geometric intersection points of the TTTTT-branes on AAA\mathscr{A}A correspond to strings in AAAAA. The flat bundle morphisms, a copy of BBB\mathscr{B}B for each geometric intersection point, dress the strings by dots of the same color. The algebra BBB\mathscr{B}B is the quotient, of the subalgebra of AAA\mathscr{A}A generated by dots, by the ideal III\mathscr{I}I of their symmetric functions.
6.2. The meaning of link homology
Since T=⨁とTeT=â¨ã¨â€ŠTeT=bigoplus_(ã¨)T_(e)T=\bigoplus_{ã¨} T_{\mathscr{e}}ã¨T=â¨ã¨Te generates DYDYD_(Y)\mathscr{D}_{Y}DY, like every Lagrangian in DYDYD_(Y)\mathscr{D}_{Y}DY, the BEUBEUBEU\mathscr{B} E \mathcal{U}BEU brane has a description as a complex
every term of which is a direct sum of TCTCT_(C)T_{\mathscr{C}}TC-branes. The complex is the projective resolution of the BEUBEUBE_(U)\mathscr{B} E_{U}BEU brane. It describes how to get the BEU∈DYBEU∈DYBE_(U)inD_(Y)\mathscr{B} E_{U} \in \mathscr{D}_{Y}BEU∈DY brane by starting with the direct sum brane
as a cohomological degree 1 and equivariant degree 0 operator, which squares to zero QA2=0QA2=0Q_(A)^(2)=0Q_{A}^{2}=0QA2=0 in the algebra AAAAA.
6.2.1.
The category of A-branes DYDYD_(Y)\mathscr{D}_{Y}DY has a second, Koszul dual set of generators, which are the vanishing cycle branes I=⨁↼I↼I=â¨â†¼â€ŠI↼I=bigoplus_(↼)I_(↼)I=\bigoplus_{\leftharpoonup} I_{\leftharpoonup}I=â¨â†¼I↼ of [2]. The IIIII - and the TTTTT-branes are dual in the sense that the only nonvanishing morphisms from the TTTTT - to the IIIII-branes are
The ICICI_(C)I_{\mathscr{C}}IC-branes and the TCTCTCT \mathcal{C}TC-branes are, respectively, the simple and the projective modules of the algebra AAAAA.
6.2.2.
Among the IIIII-branes, we find the branes Iu∈DYIu∈DYIu inD_(Y)I u \in \mathscr{D}_{Y}Iu∈DY which serve as cups. This is a wonderful simplification because it implies that from the complex in (6.5), we get for free a complex of vector spaces:
(6.9)0→homA(BE0(T),Iu{d→})→t0homA(BE1(T),Iu{d→})→t1⋯(6.9)0→homAâ¡BE0(T),Iu{d→}→t0homAâ¡BE1(T),Iu{d→}→t1⋯{:(6.9)0rarrhom_(A)(BE_(0)(T),Iu{( vec(d))})rarr"t_(0)"hom_(A)(BE_(1)(T),Iu{( vec(d))})rarr"t_(1)"cdots:}\begin{equation*}
0 \rightarrow \operatorname{hom}_{A}\left(\mathscr{B} E_{0}(T), I u\{\vec{d}\}\right) \xrightarrow{t_{0}} \operatorname{hom}_{A}\left(\mathscr{B} E_{1}(T), I u\{\vec{d}\}\right) \xrightarrow{t_{1}} \cdots \tag{6.9}
\end{equation*}(6.9)0→homAâ¡(BE0(T),Iu{d→})→t0homAâ¡(BE1(T),Iu{d→})→t1⋯
The maps in the complex (6.9) are induced from the complex in (6.5). The cohomologies of this complex are the link homologies we are after,
We learn that link homology captures only a small part of the geometry of BEUBEUBEU\mathscr{B} E \mathcal{U}BEU, the braided cup brane, or more precisely, of the complex that resolves it. Because the TTTTT branes are dual to the IIIII-branes by (6.8), almost all terms in the complex (6.9) vanish. The cohomology (6.10) of small complex that remains is the Uq(Lg)UqLgU_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(LÂ g) link homology.
6.2.4.
The complex (6.9) itself has a geometric interpretation as the Floer complex,
CF∗,∗(BEU,IU)CF∗,∗(BEU,IU)CF^(**,**)(BEU,IU)\mathrm{CF}^{*, *}(\mathscr{B} E \mathcal{U}, I U)CF∗,∗(BEU,IU)
Namely, the vector space at the kkkkk th term of the complex
is identified, by construction described in section 6.3, with that spanned by the intersection points of the BEuBEuBEu\mathscr{B} E uBEu brane and the IUIUIUI UIU brane, of cohomological degree [k][k][k][k][k] and equivariant degree {d→}{d→}{ vec(d)}\{\vec{d}\}{d→}.
encode the action of the Floer differential. A priori, computing these requires counting holomorphic disk instantons. In our case, mirror symmetry (6.2) has summed them up.
6.3. Projective resolutions from geometry
The projective resolution in (6.5) encodes all the Uq(Lg)UqLgU_(q)(^(L)g)U_{\mathcal{q}}\left({ }^{L} \mathfrak{g}\right)Uq(Lg) link homology, and more. Finding the resolution requires solving two problems, both in general difficult. We will solve simultaneously [5][5][5][5][5].
6.3.1.
The first problem is to compute which module of the algebra AAAAA the brane BEuBEuBEu\mathscr{B} E uBEu gets mapped to by the Yoneda functor
This functor, which is the source of the equivalence DY≅DADY≅DAD_(Y)~=D_(A)\mathscr{D}_{Y} \cong \mathscr{D}_{A}DY≅DA, maps a brane LLLLL to a right module for AAAAA, on which the algebra acts as
Evaluating this action requires counting disk instantons.
6.3.2.
The second problem is to find the resolution of this module, as in (6.5). The Yoneda functor maps the TeTeT_(e)T_{e}Te branes to projective modules of the algebra AAAAA, so the resolution in (6.5) is a projective resolution of the AAAAA module corresponding to the BEUBEUBE_(U)\mathscr{B} E_{\mathcal{U}}BEU brane. This problem is known to be solvable, however, only formally so, by infinite bar resolutions.
6.3.3.
In our setting, these two problems get solved together. Fortune smiles since the BEU∈DYBEU∈DYBE_(U)inD_(Y)\mathscr{B} E_{\mathcal{U}} \in \mathscr{D}_{Y}BEU∈DY branes are products of ddddd one-dimensional Lagrangians on AAA\mathcal{A}A, for which the complex resolving brane (6.5) can be deduced explicitly, from the geometry of the brane.
6.3.4.
Take a pair of branes L′L′L^(')L^{\prime}L′ and L′′L′′L^('')L^{\prime \prime}L′′ on YYYYY which are products of ddddd one-dimensional Lagrangians on AAA\mathcal{A}A, chosen to coincide up to one of their factors. Up to permutation, we can take
The Lagrangian is a cone over the intersection point PPP\mathcal{P}P of L′L′L^(')L^{\prime}L′ and L′′L′′L^('')L^{\prime \prime}L′′ which is of the form
and which also has Maslov index zero, L=Cone(P)L=Coneâ¡(P)L=Cone(P)L=\operatorname{Cone}(\mathcal{P})L=Coneâ¡(P).
Conversely, any LLLLL brane which is of the product form in (6.11) can be written as a complex [11][11][11][11][11]
(6.13)L≅L′→PL′′(6.13)L≅L′→PL′′{:(6.13)L~=L^(')rarr"P"L^(''):}\begin{equation*}
L \cong L^{\prime} \xrightarrow{\mathcal{P}} L^{\prime \prime} \tag{6.13}
\end{equation*}(6.13)L≅L′→PL′′
with an explicit map PPP\mathcal{P}P coming from a one-dimensional intersection point in one of its factors, as in (6.12).
6.3.5.
To find the projective resolution of the BEUBEUBEU\mathscr{B} E \mathcal{U}BEU brane in (6.5), start by isotoping the brane, by stretching it straight along the cylinder.
Let the brane break at the two infinities of AAA\mathscr{A}A, to get the direct sum brane BE(T)BE(T)BE(T)\mathscr{B} E(T)BE(T) in (6.6), on which the complex is based. To find the maps in the complex, record how the brane breaks, iterating the above construction, one one-dimensional intersection point at the time. Every intersection point of the form (6.12) translates to a specific element of the algebra AAAAA and a specific map in the complex. The result is a product of ddddd one-dimensional complexes, which describes factors of BEUBEUBEU\mathscr{B} E \mathcal{U}BEU, and captures almost all the terms in the differential QAQAQ_(A)Q_{A}QA. The remaining ones follow, up to quasi-isomorphisms, by asking that the differential closes QA2=0QA2=0Q_(A)^(2)=0Q_{A}^{2}=0QA2=0 in the algebra AAAAA. The fact that not all terms in QAQAQ_(A)Q_{A}QA are geometric is a general feature of d>1d>1d > 1d>1d>1 theories.
In practice, it is convenient to first break the brane one of the two infinities of AAA\mathcal{A}A, and only then on the other. The branes at the intermediate stage are images, under the h∗h∗h^(**)h^{*}h∗ functor, of stable basis branes [7,61][7,61][7,61][7,61][7,61] on DXDXDX\mathscr{D} XDX. The stable basis branes play a similar role to that of Verma modules in category OOO\mathcal{O}O. The detailed algorithm is given in [5].
6.3.6.
As an example, take the left-handed trefoil and Lg=su2Lg=su2^(L)g=su_(2){ }^{L} \mathfrak{g}=\mathfrak{s u}_{2}Lg=su2, which leads to the brane configuration from Figure 9. For simplicity, consider the reduced knot homology, where the unknot homology is set to be trivial. As in Heegard-Floer theory, this corresponds to erasing a component from the BEuBEuBEu\mathscr{B} E uBEu and the IuIuIuI uIu branes, and leads to Figure 10. This also brings us back to the setting of our running example, where YYYYY is the equivariant mirror to XXX\mathcal{X}X, the resolution of the An−1An−1A_(n-1)A_{n-1}An−1 surface singularity, with n=4n=4n=4n=4n=4.
FIGURE 10
Resolution of the BEUBEUBEU\mathscr{B} E \mathcal{U}BEU brane corresponding to the reduced trefoil. The axis of the cylinder AAA\mathscr{A}A is oriented vertically here; the branes do not wind around the S1S1S^(1)S^{1}S1.
The corresponding algebra A=⨁i,j=0n−1HomDY∗(Ti,Tj)A=â¨i,j=0n−1 HomDY∗â¡Ti,TjA=bigoplus_(i,j=0)^(n-1)Hom_(D_(Y))^(**)(T_(i),T_(j))A=\bigoplus_{i, j=0}^{n-1} \operatorname{Hom}_{\mathscr{D}_{Y}}^{*}\left(T_{i}, T_{j}\right)A=â¨i,j=0n−1HomDY∗â¡(Ti,Tj) is the path algebra of an affine An−1An−1A_(n-1)A_{n-1}An−1 quiver, whose nodes correspond to TiTiT_(i)T_{i}Ti branes. The arrows ai+1,i∈ai+1,i∈a_(i+1,i)ina_{i+1, i} \inai+1,i∈HomDY(Ti,Ti+1)HomDYâ¡Ti,Ti+1Hom_(D_(Y))(T_(i),T_(i+1))\operatorname{Hom}_{\mathscr{D}_{Y}}\left(T_{i}, T_{i+1}\right)HomDYâ¡(Ti,Ti+1) and bi,i+1∈HomDY(Ti+1,Ti{1})bi,i+1∈HomDYâ¡Ti+1,Ti{1}b_(i,i+1)inHom_(DY)(T_(i+1),T_(i){1})b_{i, i+1} \in \operatorname{Hom}_{\mathscr{D} Y}\left(T_{i+1}, T_{i}\{1\}\right)bi,i+1∈HomDYâ¡(Ti+1,Ti{1}) satisfy ai,i−1bi−1,i=0=bi,i+1ai+1,iai,i−1bi−1,i=0=bi,i+1ai+1,ia_(i,i-1)b_(i-1,i)=0=b_(i,i+1)a_(i+1,i)a_{i, i-1} b_{i-1, i}=0=b_{i, i+1} a_{i+1, i}ai,i−1bi−1,i=0=bi,i+1ai+1,i, with iiiii defined modulo nnnnn. The aaaaa 's and bbbbb 's correspond to intersections of TTTTT-branes, near one or the other infinity of AAA\mathscr{A}A; we have suppressed their ΛΛLambda\LambdaΛ-equivariant degrees.
Isotope the BEuBEuBEu\mathscr{B} E \mathcal{u}BEu brane straight along the cylinder AAA\mathcal{A}A. Let it break into TTTTT-branes, as in Figure 10, while recording how the brane breaks, one connected sum at a time. Every connected sum of a pair of TTTTT-branes is a cone over their intersection point, at one of the two infinities of AAA\mathcal{A}A, and a specific element of the algebra AAAAA. This leads to the complex
which closes by the AAAAA-algebra relations.
The reduced homology of the trefoil is the cohomology of the complex homA(BE∙,Iu{d})homAâ¡BE∙,Iâ¡u{d}hom_(A)(BE^(∙),Iu{d})\operatorname{hom}_{A}\left(\mathscr{B} E^{\bullet}, \operatorname{I} u\{d\}\right)homAâ¡(BE∙,Iâ¡u{d}) in (6.9). The only non-zero contributions come from the T2T2T_(2)T_{2}T2 brane, since the cup brane IU=I2IU=I2IU=I_(2)I U=I_{2}IU=I2 is dual to it. All the maps evaluate to zero, as IUIUIUI UIU brane is a simple module for AAAAA. As a consequence,
HomDY(BEu,Iu[k]{d})=Hk(homA(BE∙,I2{d}))HomDYâ¡(BEu,Iu[k]{d})=HkhomAâ¡BE∙,I2{d}Hom_(D_(Y))(BEu,Iu[k]{d})=H^(k)(hom_(A)(BE^(∙),I_(2){d}))\operatorname{Hom}_{\mathscr{D _ { Y }}}(\mathscr{B} E u, I u[k]\{d\})=H^{k}\left(\operatorname{hom}_{A}\left(\mathscr{B} E^{\bullet}, I_{2}\{d\}\right)\right)HomDYâ¡(BEu,Iu[k]{d})=Hk(homAâ¡(BE∙,I2{d}))
equals to ZZZ\mathbb{Z}Z only for (k,d)=(0,0),(2,−2),(3,−3)(k,d)=(0,0),(2,−2),(3,−3)(k,d)=(0,0),(2,-2),(3,-3)(k, d)=(0,0),(2,-2),(3,-3)(k,d)=(0,0),(2,−2),(3,−3), and vanishes otherwise. Here, k=Mk=Mk=Mk=Mk=M is the Maslov or cohomological degree and d=Jd=Jd=Jd=Jd=J the Jones grading. This is the reduced Khovanov homology of the left-handed trefoil, up to regrading: Khovanov's (i,j)(i,j)(i,j)(i, j)(i,j) gradings are related to (M,J)(M,J)(M,J)(M, J)(M,J) by i=M+2J+i0i=M+2J+i0i=M+2J+i_(0)i=M+2 J+i_{0}i=M+2J+i0 and j=2J+j0j=2J+j0j=2J+j_(0)j=2 J+j_{0}j=2J+j0 where i0=0,j0=d+n+−i0=0,j0=d+n+−i_(0)=0,j_(0)=d+n_(+)-i_{0}=0, j_{0}=d+n_{+}-i0=0,j0=d+n+−n−n−n_(-)n_{-}n−, where n+=0,n−=3n+=0,n−=3n_(+)=0,n_(-)=3n_{+}=0, n_{-}=3n+=0,n−=3 are the numbers of positive and negative crossings, and d=1d=1d=1d=1d=1 is the dimension of YYYYY [2].
6.3.7.
The theory extends to non-simply-laced Lie algebras, and to Lie superalgebras glm∣nglm∣ngl_(m∣n)\mathfrak{g l}_{m \mid n}glm∣n and m∣2nm∣2n_(m∣2n)\mathfrak{}_{m \mid 2 n}m∣2n, as described in [5]. The algebra AAAAA corresponding to LgLg^(L)g{ }^{L} \mathrm{~g}L g which is a Lie superalgebra, is not an ordinary associative algebra but a differential graded algebra; the projective resolutions are then in terms of twisted complexes. This section gives a method for solving the theory which is new even for Lg=gl1∣1Lg=gl1∣1^(L)g=gl_(1∣1){ }^{L} \mathfrak{g}=\mathfrak{g l}_{1 \mid 1}Lg=gl1∣1, corresponding to Heegard-Floer theory. The solution differs from that in [65], in particular since our Heegard surface is A=R×S1A=R×S1A=RxxS^(1)\mathcal{A}=\mathbb{R} \times S^{1}A=R×S1, independent of the link.
ACKNOWLEDGMENTS
This work grew out of earlier collaborations with Andrei Okounkov, which were indispensable. It includes results obtained jointly with Ivan Danilenko, Elise LePage, Yixuan Li, Michael McBreen, Miroslav Rapcak, Vivek Shende, and Peng Zhou. I am grateful to all of them for collaboration. I was supported by the NSF foundation grant PHY1820912, by the Simons Investigator Award, and by the Berkeley Center for Theoretical Physics.
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MINA AGANAGIC
Department of Mathematics, University of California, Berkeley, USA and Center for Theoretical Physics, University of California, Berkeley, USA, aganagic @ berkeley.edu
VECTOR BUNDLES ON ALGEBRAIC VARIETIES
ARAVIND ASOK AND JEAN FASEL
ABSTRACT
We survey recent developments related to the problem of classifying vector bundles on algebraic varieties. We focus on the striking analogies between topology and algebraic geometry, and the way in which the Morel-Voevodsky motivic homotopy category can be used to exploit those analogies.
In the mid-1950s, Serre created a dictionary between the theory of vector bundles in topology and the theory of projective modules over a commutative ring [55, 56]. Echoing M. M. Postnikov's MathSciNet review of Serre's paper, J. F. Adams prosaically wrote in his review of H. Bass' paper [22]: "This leads to the following programme: take definitions, constructions and theorems from bundle-theory; express them as particular cases of definitions, constructions and statements about finitely-generated projective modules over a general ring; and finally, try to prove the statements under suitable assumptions". One of the results Serre presented to illustrate this dictionary was the algebro-geometric analog of existence of nowhere vanishing sections for negative corank projective modules, now frequently referred to as Serre's splitting theorem, which we recall in algebro-geometric formulation: if EEE\mathscr{E}E is a rank rrrrr vector bundle over a Noetherian affine scheme XXXXX of dimension ddddd, then when r>d,E≅E′⊕OXr>d,E≅E′⊕OXr > d,E~=E^(')o+O_(X)r>d, \mathcal{E} \cong \mathcal{E}^{\prime} \oplus \mathcal{O}_{X}r>d,E≅E′⊕OX.
After the Pontryagin-Steenrod representability theorem, topological vector bundles on smooth manifolds (or spaces having the homotopy type of a CW complex) can be analyzed using homotopy theoretic techniques. Extending Serre's analogy further and using celebrated work of Bass, Quillen, Suslin, and Lindel, F. Morel showed that algebraic vector bundles on smooth affine varieties could be studied using an algebro-geometric homotopy theory: the Morel-Voevodsky motivic homotopy theory. In this note, we survey recent developments in the theory of algebraic vector bundles motivated by this circle of ideas, making sure to indicate the striking analogies between topology and algebraic geometry.
To give the reader a taste of the methods we will use, we mention two results here.
First, we state an improvement of Serre's splitting theorem mentioned above (for the moment it suffices to know that A1A1A^(1)\mathbb{A}^{1}A1-cohomological dimension is bounded above by Krull dimension, but can be strictly smaller). Second, we will discuss the splitting problem for projective modules in corank 1, which goes beyond any classical results.
Theorem 1.1. If kkkkk is a field, and XXXXX is a smooth affine kkkkk-scheme of A1A1A^(1)\mathbb{A}^{1}A1-cohomological dimension ≤d≤d<= d\leq d≤d, then any rank r>dr>dr > dr>dr>d bundle splits off a trivial rank 1 summand.
Conjecture 1.2. Assume kkkkk is an algebraically closed field, and X=SpecRX=Specâ¡RX=Spec RX=\operatorname{Spec} RX=Specâ¡R is a smooth affine kkkkk-variety of dimension ddddd. A rank d−1d−1d-1d-1d−1 vector bundle &&&\mathcal{\&}& on XXXXX splits off a free rank 1 summand if and only if 0=cd−1(E)∈CHd−1(X)0=cd−1(E)∈CHd−1(X)0=c_(d-1)(E)inCH^(d-1)(X)0=c_{d-1}(\mathcal{E}) \in \mathrm{CH}^{d-1}(X)0=cd−1(E)∈CHd−1(X).
In Theorem 4.12 we verify Conjecture 1.2 in case d=3,4d=3,4d=3,4d=3,4d=3,4 (and kkkkk has characteristic not equal to 2). To motivate the techniques used to establish these results, we begin by analyzing topological variants of these conjectures. We close this note with a discussion of joint work with Mike Hopkins which addresses the difficult problem of constructing interesting low rank vector bundles on "simple" algebraic varieties. As with any survey, this one reflects the biases and knowledge of the authors. Limitations of space have prevented us from talking about a number of very exciting and closely related topics.
2. A FEW TOPOLOGICAL STORIES
In this section, we recall a few topological constructions that elucidate the approaches we use to analyze corresponding algebro-geometric questions studied later.
2.1. Moore-Postnikov factorizations
Suppose f:E→Bf:E→Bf:E rarr Bf: E \rightarrow Bf:E→B is a morphism of pointed, connected topological spaces having the homotopy type of CWCWCW\mathrm{CW}CW complexes that induces an isomorphism of fundamental groups (for simplicity of discussion). Write FFFFF for the "homotopy" fiber of fffff, so that there is a fiber sequence
F→E→fBF→E→fBF rarr Erarr"f"BF \rightarrow E \xrightarrow{f} BF→E→fB
yielding a long exact sequence relating the homotopy of F,EF,EF,EF, EF,E, and BBBBB.
A basic question that arises repeatedly is the following: given a map M→BM→BM rarr BM \rightarrow BM→B, when can it be lifted along fffff to a map M→EM→EM rarr EM \rightarrow EM→E ? To approach this problem, one method is to factor fffff in such a way as to break the original lifting problem into simpler problems where existence of a lift can be checked by, say, cohomological means.
One systematic approach to analyzing this question was laid out in the work of Moore-Postnikov. In this case, one factors fffff so as to build EEEEE out of BBBBB by sequentially adding higher homotopy of fffff (keeping track of the induced action of π1(E)≅π1(B)Ï€1(E)≅π1(B)pi_(1)(E)~=pi_(1)(B)\pi_{1}(E) \cong \pi_{1}(B)Ï€1(E)≅π1(B) on the fiber). In more detail, the Moore-Postnikov tower of fffff consists of a sequence of spaces τ≤if,i≥0τ≤if,i≥0tau_( <= i)f,i >= 0\tau_{\leq i} f, i \geq 0τ≤if,i≥0 and morphisms fitting into the following diagram:
The key properties of this factorization are that (i) the composite maps E→τ≤if→BE→τ≤if→BE rarrtau_( <= i)f rarr BE \rightarrow \tau_{\leq i} f \rightarrow BE→τ≤if→B all coincide with fffff, (ii) the maps E→τ≤ifE→τ≤ifE rarrtau_( <= i)fE \rightarrow \tau_{\leq i} fE→τ≤if induce isomorphisms on homotopy groups in degrees ≤i≤i<= i\leq i≤i, (iii) the maps τ≤if→Bτ≤if→Btau_( <= i)f rarr B\tau_{\leq i} f \rightarrow Bτ≤if→B induce isomorphisms on homotopy in degrees >i+1>i+1> i+1>i+1>i+1, and (iv) there is a homotopy pullback diagram of the form
In particular, the morphism τ≤if→τ≤i−1fτ≤if→τ≤i−1ftau_( <= i)f rarrtau_( <= i-1)f\tau_{\leq i} f \rightarrow \tau_{\leq i-1} fτ≤if→τ≤i−1f is a twisted principal fibration, which means that a morphism M→τ≤i−1fM→τ≤i−1fM rarrtau_( <= i-1)fM \rightarrow \tau_{\leq i-1} fM→τ≤i−1f lifts along the tower if and only if the composite M→Kπ1(E)(πi(F),i+1)M→KÏ€1(E)Ï€i(F),i+1M rarrK^(pi_(1)(E))(pi_(i)(F),i+1)M \rightarrow K^{\pi_{1}(E)}\left(\pi_{i}(F), i+1\right)M→KÏ€1(E)(Ï€i(F),i+1) lifts to Bπ1(E)BÏ€1(E)Bpi_(1)(E)\mathrm{B} \pi_{1}(E)BÏ€1(E). The latter map amounts to a cohomology class on MMMMM with coefficients in a local coefficient system; this cohomology class is pulled back from a "universal example" the kkkkk-invariant at the corresponding stage. If the obstruction vanishes, a lift exists. Lifts are not unique in general, but the ambiguity in choice of a lift can also be described.
2.2. The topological splitting problem
In this section, to motivate some of the algebro-geometric results we will describe later, we review the problem of deciding whether a bundle of corank 0 or 1 on a closed smooth manifold MMMMM of dimension d+1d+1d+1d+1d+1 has a nowhere vanishing section. We now phrase this problem as a lifting problem of the type described in the preceding section.
In this case, the relevant lifting problem is:
To analyze the lifting problem, we describe the Moore-Postnikov factorization of fffff. The homotopy fiber of fffff coincides with the standard sphere Sd−1≅O(d)/O(d−1)Sd−1≅O(d)/O(d−1)S^(d-1)~=O(d)//O(d-1)S^{d-1} \cong O(d) / O(d-1)Sd−1≅O(d)/O(d−1).
The stabilization map O(d−1)→O(d)O(d−1)→O(d)O(d-1)rarr O(d)O(d-1) \rightarrow O(d)O(d−1)→O(d) is compatible with the determinant, and there are thus induced isomorphisms π1(BO(d−1))→π1(BO(d))≅Z/2Ï€1(BO(d−1))→π1(BO(d))≅Z/2pi_(1)(BO(d-1))rarrpi_(1)(BO(d))~=Z//2\pi_{1}(\mathrm{~B} O(d-1)) \rightarrow \pi_{1}(\mathrm{~B} O(d)) \cong \mathbb{Z} / 2Ï€1( BO(d−1))→π1( BO(d))≅Z/2 compatible with fffff. Note, however, that the action of Z/2Z/2Z//2\mathbb{Z} / 2Z/2 on the higher homotopy of BO(d)BO(d)BO(d)\mathrm{B} O(d)BO(d) depends on the parity of ddddd : when ddddd is odd the action is trivial, while if ddddd is even the action is nontrivial in general and even fails to be nilpotent. Of course, Sd−1Sd−1S^(d-1)S^{d-1}Sd−1 is (d−2)(d−2)(d-2)(d-2)(d−2)-connected.
Remark 2.1. At this stage, the fact that bundles of negative corank on spaces have the homotopy type of a CW complex of dimension ddddd follows immediately from obstruction theory granted the assertion that the sphere SrSrS^(r)S^{r}Sr is an (r−1)(r−1)(r-1)(r-1)(r−1)-connected space in conjunction with the fact that negative corank means r>dr>dr > dr>dr>d.
In order to write down obstructions, we need some information about the homotopy of spheres: the first nonvanishing homotopy group of Sd−1Sd−1S^(d-1)S^{d-1}Sd−1 is πd−1(Sd−1)Ï€d−1Sd−1pi_(d-1)(S^(d-1))\pi_{d-1}\left(S^{d-1}\right)Ï€d−1(Sd−1) which coincides
with ZZZ\mathbb{Z}Z for all d≥2d≥2d >= 2d \geq 2d≥2 (via the degree map). Likewise, πd(Sd−1)Ï€dSd−1pi_(d)(S^(d-1))\pi_{d}\left(S^{d-1}\right)Ï€d(Sd−1) is ZZZ\mathbb{Z}Z if d=3d=3d=3d=3d=3 and Z/2Z/2Z//2\mathbb{Z} / 2Z/2 if d>3d>3d > 3d>3d>3 and is generated by a suitable suspension of the classical Hopf map η:S3→S2η:S3→S2eta:S^(3)rarrS^(2)\eta: S^{3} \rightarrow S^{2}η:S3→S2.
Assume now XXXXX is a space having the homotopy type of a finite CWCWCW\mathrm{CW}CW complex of dimension d+1d+1d+1d+1d+1 for some fixed integer d≥2d≥2d >= 2d \geq 2d≥2 (to eliminate some uninteresting cases) and ξ:X→BO(d)ξ:X→BO(d)xi:X rarrBO(d)\xi: X \rightarrow \mathrm{B} O(d)ξ:X→BO(d) classifies a rank ddddd vector bundle on XXXXX. The first nonzero kkkkk-invariant for fffff yields a map X→KZ/2(πd−1(Sd−1),d)X→KZ/2Ï€d−1Sd−1,dX rarrK^(Z//2)(pi_(d-1)(S^(d-1)),d)X \rightarrow K^{\mathbb{Z} / 2}\left(\pi_{d-1}\left(S^{d-1}\right), d\right)X→KZ/2(Ï€d−1(Sd−1),d), i.e., an element
called the (twisted) Euler class, where Z[σ]Z[σ]Z[sigma]\mathbb{Z}[\sigma]Z[σ] is ZZZ\mathbb{Z}Z twisted by the orientation character σσsigma\sigmaσ defined by applying π1Ï€1pi_(1)\pi_{1}Ï€1 to the morphism X→BO(d)→B(Z/2)X→BO(d)→B(Z/2)X rarrBO(d)rarrB(Z//2)X \rightarrow \mathrm{B} O(d) \rightarrow \mathrm{B}(\mathbb{Z} / 2)X→BO(d)→B(Z/2).
Assuming this primary obstruction vanishes, one may choose a lift to the next stage of the Postnikov tower. If we fix a lift, then there is a well-defined secondary obstruction to lifting, that comes from the next kkkkk-invariant: this obstruction is given by a map X→KZ/2(πd(Sd−1),d+1)X→KZ/2Ï€dSd−1,d+1X rarrK^(Z//2)(pi_(d)(S^(d-1)),d+1)X \rightarrow K^{\mathbb{Z} / 2}\left(\pi_{d}\left(S^{d-1}\right), d+1\right)X→KZ/2(Ï€d(Sd−1),d+1), i.e., a cohomology class in Hd+1(X,Z[σ])Hd+1(X,Z[σ])H^(d+1)(X,Z[sigma])H^{d+1}(X, \mathbb{Z}[\sigma])Hd+1(X,Z[σ]) if d=3d=3d=3d=3d=3 or Hd+1(X,Z/2)Hd+1(X,Z/2)H^(d+1)(X,Z//2)H^{d+1}(X, \mathbb{Z} / 2)Hd+1(X,Z/2) if d≠3d≠3d!=3d \neq 3d≠3; in the latter case the choice of orientation character no longer affects this cohomology group.
If one tracks the effect of choice of lift on the obstruction class described above, one obtains a map KZ/2(πd−1(Sd−1),d−1)→KZ/2(πd(Sd−1),d+1)KZ/2Ï€d−1Sd−1,d−1→KZ/2Ï€dSd−1,d+1K^(Z//2)(pi_(d-1)(S^(d-1)),d-1)rarrK^(Z//2)(pi_(d)(S^(d-1)),d+1)K^{\mathbb{Z} / 2}\left(\pi_{d-1}\left(S^{d-1}\right), d-1\right) \rightarrow K^{\mathbb{Z} / 2}\left(\pi_{d}\left(S^{d-1}\right), d+1\right)KZ/2(Ï€d−1(Sd−1),d−1)→KZ/2(Ï€d(Sd−1),d+1), which is a twisted cohomology operation. If d=3d=3d=3d=3d=3, the map in question is a twisted version of the Pontryagin squaring operation, while if d>3d>3d > 3d>3d>3 the operation can be described as Sq2+w2∪Sq2+w2∪Sq^(2)+w_(2)uu\mathrm{Sq}^{2}+w_{2} \cupSq2+w2∪, where w2w2w_(2)w_{2}w2 is the second Stiefel-Whitney class of the bundle. In that case, the secondary obstruction yields a well-defined coset in
This description of the primary and secondary obstructions was laid out carefully by the early 1950s by S. D. Liao [37].
Finally, the dimension assumption on XXXXX guarantees that a lift of ξξxi\xiξ along fffff exists if and only if these two obstructions vanish. In principle, this kind of analysis can be continued, though the calculations become more involved as the indeterminacy created by successive choices of lifts becomes harder to control and information about higher unstable homotopy of spheres is also harder to obtain. For a thorough treatment of this and even more general situations, we refer the reader to [61].
Remark 2.2. The analysis of the obstructions can be improved by organizing the calculations differently. The Moore-Postnikov factorization has the effect of factoring a map f:X→Yf:X→Yf:X rarr Yf: X \rightarrow Yf:X→Y as a tower of fibrations where the relevant fibers are Eilenberg-Mac Lane spaces. However, there are many other ways to produce factorizations of fffff with different constraints on the "cohomological" properties of pieces of the tower.
3. A QUICK REVIEW OF MOTIVIC HOMOTOPY THEORY
Motivic homotopy theory, introduced by F. Morel and V. Voevodsky [41], provides a homotopy theory for schemes over a base. While there are a number of different approaches
to constructing the motivic homotopy category that work in great generality, we work in a very concrete situation. By an algebraic variety over a field kkkkk, we will mean a separated, finite type, reduced kkkkk-scheme. We write SmkSmkSm_(k)\mathrm{Sm}_{k}Smk for the category of smooth algebraic varieties; for later use, we will also write Smkaff Smkaff Sm_(k)^("aff ")\mathrm{Sm}_{k}^{\text {aff }}Smkaff for the full subcategory of SmkSmkSm_(k)\mathrm{Sm}_{k}Smk consisting of affine schemes.
The category SmkSmkSm_(k)\mathrm{Sm}_{k}Smk is "too small" to do homotopy theory, in the sense that various natural categorical constructions one would like to make (increasing unions, quotients by subspaces, etc.) can leave the category. As such, one first enlarges SmkSmkSm_(k)\mathrm{Sm}_{k}Smk to a suitable category SpckSpckSpc_(k)\mathrm{Spc}_{k}Spck of "spaces"; one may take SpckSpckSpc_(k)\mathrm{Spc}_{k}Spck to be the category of simplicial presheaves on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk and the functor Smk→SpckSmk→SpckSm_(k)rarrSpc_(k)\mathrm{Sm}_{k} \rightarrow \mathrm{Spc}_{k}Smk→Spck is given by the Yoneda embedding followed by the functor viewing a presheaf on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk as a constant simplicial presheaf.
3.1. Homotopical sheaf theory
Passing to SpckSpckSpc_(k)\mathrm{Spc}_{k}Spck has the effect of destroying certain colimits that one would like to retain. To recover the colimits that have been lost, one localizes SpckSpckSpc_(k)\mathrm{Spc}_{k}Spck and passes to a suitable "local" homotopy category of the sort first studied in detail by K. Brown-S. Gersten, A. Joyal, and J.F. Jardine: one fixes a Grothendieck topology τÏ„tau\tauÏ„ on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk and inverts the so-called τÏ„tau\tauÏ„-local weak equivalences on SpckSpckSpc_(k)\mathrm{Spc}_{k}Spck; we refer the reader to [34] for a textbook treatment. We write Hτ(k)HÏ„(k)H_(tau)(k)\mathrm{H}_{\tau}(k)HÏ„(k) for the resulting localization of SpckSpckSpc_(k)\mathrm{Spc}_{k}Spck. If X∈SpckX∈SpckXinSpc_(k)\mathscr{X} \in \mathrm{Spc}_{k}X∈Spck, then a base-point for XXX\mathscr{X}X is a morphism x:Speck→Xx:Specâ¡k→Xx:Spec k rarrXx: \operatorname{Spec} k \rightarrow \mathscr{X}x:Specâ¡k→X splitting the structure morphism. There is an associated pointed homotopy category and these homotopy categories can be thought of as providing a convenient framework for "nonabelian" homological algebra.
In the category of pointed spaces, we can make sense of wedge sums and smash products, just as in ordinary topology. We also define spheres Si,i≥0Si,i≥0S^(i),i >= 0S^{i}, i \geq 0Si,i≥0, as the constant simplicial presheaves corresponding to the simplicial sets SiSiS^(i)S^{i}Si. For any pointed space (X,x)(X,x)(X,x)(\mathscr{X}, x)(X,x), we define its homotopy sheaves πi(X,x)Ï€i(X,x)pi_(i)(X,x)\pi_{i}(\mathscr{X}, x)Ï€i(X,x) as the Nisnevich sheaves associated with the presheaves on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk defined by
here the subscript + means adjoint a disjoint base-point. These homotopy sheaves may be used to formulate a Whitehead theorem.
If GGG\mathbf{G}G is a Nisnevich sheaf of groups on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk, then there is a classifying space B G such that for any smooth kkkkk-scheme XXXXX one has a functorial identification of pointed sets of the form
where we as usual identify isomorphism classes of ranknrankâ¡nrank n\operatorname{rank} nrankâ¡n vector bundles locally trivial with respect to the Zariski topology on XXXXX with GLnGLnGL_(n)\mathrm{GL}_{n}GLn-torsors (and the choice of topology does not matter).
If AAA\mathbf{A}A is any Nisnevich sheaf of abelian groups on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk, then for any integer n≥0n≥0n >= 0n \geq 0n≥0 there are Eilenberg-Mac Lane spaces K(A,n)K(A,n)K(A,n)\mathrm{K}(\mathbf{A}, n)K(A,n), i.e., spaces with exactly one nonvanishing homotopy sheaf, appearing in degree nnnnn, isomorphic to AAA\mathbf{A}A. For such spaces, homHNis (k)(X,K(A,n))homHNis (k)â¡(X,K(A,n))hom_(H_("Nis ")(k))(X,K(A,n))\operatorname{hom}_{\mathrm{H}_{\text {Nis }}(k)}(X, \mathrm{~K}(\mathbf{A}, n))homHNis (k)â¡(X, K(A,n)) has a natural abelian group structure, and there are functorial isomorphisms of abelian groups
With this definition, for essentially formal reasons there is a suspension isomorphism for Nisnevich cohomology with respect to the suspension S1∧(−)S1∧(−)S^(1)^^(-)S^{1} \wedge(-)S1∧(−).
3.2. The motivic homotopy category
The motivic homotopy category is obtained as a further localization of HNis(k)HNis(k)H_(Nis)(k)\mathrm{H}_{\mathrm{Nis}}(k)HNis(k) : one localizes at the projection morphisms X×A1→XX×A1→XXxxA^(1)rarrX\mathscr{X} \times \mathbb{A}^{1} \rightarrow \mathscr{X}X×A1→X. We write H(k)H(k)H(k)\mathrm{H}(k)H(k) for the resulting homotopy category; isomorphisms in this category will be referred to as A1A1A^(1)\mathbb{A}^{1}A1-weak equivalences. Following the notation in classical homotopy theory, we write
and refer to this set as the set of A1A1A^(1)\mathbb{A}^{1}A1-homotopy classes of maps from XXX\mathscr{X}X to YYY\mathscr{Y}Y.
If XXX\mathscr{X}X is a space, we will write π0A1(X)Ï€0A1(X)pi_(0)^(A^(1))(X)\pi_{0}^{\mathbb{A}^{1}}(\mathscr{X})Ï€0A1(X) for the Nisnevich sheaf associated with the presheaf U↦[U,X]A1U↦[U,X]A1U|->[U,X]_(A^(1))U \mapsto[U, \mathscr{X}]_{\mathbb{A}^{1}}U↦[U,X]A1 on SmkSmkSm_(k)\operatorname{Sm}_{k}Smk; we refer to π0A1(X)Ï€0A1(X)pi_(0)^(A^(1))(X)\pi_{0}^{\mathbb{A}^{1}}(\mathscr{X})Ï€0A1(X) as the sheaf of connected components, and we say that XXX\mathscr{X}X is A1A1A^(1)\mathbb{A}^{1}A1-connected if π0A1(X)Ï€0A1(X)pi_(0)^(A^(1))(X)\pi_{0}^{\mathbb{A}^{1}}(\mathscr{X})Ï€0A1(X) is the sheaf Spec(k)Specâ¡(k)Spec(k)\operatorname{Spec}(k)Specâ¡(k).
We consider GmGmG_(m)\mathbb{G}_{m}Gm as a pointed space, with base point its identity section 1 . In that case, we define motivic spheres
We caution the reader that there are a number of different indexing conventions used for motivic spheres. One defines bigraded homotopy sheaves πi,jA1(X,x)Ï€i,jA1(X,x)pi_(i,j)^(A^(1))(X,x)\pi_{i, j}^{\mathbb{A}^{1}}(\mathscr{X}, x)Ï€i,jA1(X,x) for any pointed space as the Nisnevich sheaves associated with the presheaves on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk
we write πiA1(X,x)Ï€iA1(X,x)pi_(i)^(A^(1))(X,x)\pi_{i}^{\mathbb{A}^{1}}(\mathscr{X}, x)Ï€iA1(X,x) for πi,0A1(X)Ï€i,0A1(X)pi_(i,0)^(A^(1))(X)\pi_{i, 0}^{\mathbb{A}^{1}}(\mathscr{X})Ï€i,0A1(X). We will say that a pointed space (X,x)(X,x)(X,x)(\mathscr{X}, x)(X,x) is A1A1A^(1)\mathbb{A}^{1}A1 - kkkkk-connected for some integer k≥1k≥1k >= 1k \geq 1k≥1 if it is A1A1A^(1)\mathbb{A}^{1}A1-connected and the sheaves πiA1(X,x)Ï€iA1(X,x)pi_(i)^(A^(1))(X,x)\pi_{i}^{\mathbb{A}^{1}}(\mathscr{X}, x)Ï€iA1(X,x) are trivial for 1≤i≤k1≤i≤k1 <= i <= k1 \leq i \leq k1≤i≤k. Because of the form of the Whitehead theorem in the Nisnevich local homotopy category, the sheaves πiA1(−)Ï€iA1(−)pi_(i)^(A^(1))(-)\pi_{i}^{\mathbb{A}^{1}}(-)Ï€iA1(−) detect A1A1A^(1)\mathbb{A}^{1}A1-weak equivalences.
We write Δk∙Δk∙Delta_(k)^(∙)\Delta_{k}^{\bullet}Δk∙ for the cosimplicial affine space with
For any space XXX\mathscr{X}X, we write SingA1XSingA1â¡XSing^(A^(1))X\operatorname{Sing}^{\mathbb{A}^{1}} \mathscr{X}SingA1â¡X for the space diaghom__(Δ∙,X)diagâ¡hom__Δ∙,Xdiag hom__(Delta^(∙),X)\operatorname{diag} \underline{\underline{h o m}}\left(\Delta^{\bullet}, \mathscr{X}\right)diagâ¡hom__(Δ∙,X). There is a canonical map X→SingA1XX→SingA1â¡XXrarrSing^(A^(1))X\mathscr{X} \rightarrow \operatorname{Sing}^{\mathbb{A}^{1}} \mathscr{X}X→SingA1â¡X and the space SingA1XSingA1â¡XSing^(A^(1))X\operatorname{Sing}^{\mathbb{A}^{1}} \mathscr{X}SingA1â¡X is called the singular construction on XXX\mathscr{X}X. For
a smooth scheme UUUUU, the set of connected components π0(SingA1X(U))Ï€0SingA1â¡X(U)pi_(0)(Sing^(A^(1))X(U))\pi_{0}\left(\operatorname{Sing}^{\mathbb{A}^{1}} \mathscr{X}(U)\right)Ï€0(SingA1â¡X(U)) will be called the set of naive A1A1A^(1)\mathbb{A}^{1}A1-homotopy classes of maps U→XU→XU rarrXU \rightarrow \mathscr{X}U→X (by construction, it is the quotient of the set of morphisms U→XU→XU rarrXU \rightarrow \mathscr{X}U→X by the equivalence relation generated by maps U×A1→XU×A1→XU xxA^(1)rarrXU \times \mathbb{A}^{1} \rightarrow \mathscr{X}U×A1→X ). Again, by definition there is a comparison morphism
(3.1)π0( Sing A1X(U))→[U,X]A1(3.1)Ï€0 Sing A1X(U)→[U,X]A1{:(3.1)pi_(0)(" Sing "^(A^(1))X(U))rarr[U","X]_(A^(1)):}\begin{equation*}
\pi_{0}\left(\text { Sing }^{\mathbb{A}^{1}} \mathscr{X}(U)\right) \rightarrow[U, \mathscr{X}]_{\mathbb{A}^{1}} \tag{3.1}
\end{equation*}(3.1)π0( Sing A1X(U))→[U,X]A1
Typically, the map (3.1) is far from being a bijection.
3.3. A1A1A^(1)\mathbb{A}^{1}A1-weak equivalences
We now give a number of examples of A1A1A^(1)\mathbb{A}^{1}A1-weak equivalences, highlighting some examples and constructions that will be important in the sequel.
Example 3.1. A smooth kkkkk-scheme XXXXX is called A1A1A^(1)\mathbb{A}^{1}A1-contractible if the structure morphism X→SpeckX→Specâ¡kX rarr Spec kX \rightarrow \operatorname{Spec} kX→Specâ¡k is an A1A1A^(1)\mathbb{A}^{1}A1-weak equivalence. By construction, AnAnA^(n)\mathbb{A}^{n}An is an A1A1A^(1)\mathbb{A}^{1}A1-contractible smooth kkkkk-scheme. However, there are a plethora of A1A1A^(1)\mathbb{A}^{1}A1-contractible smooth kkkkk-schemes that are nonisomorphic to AnAnA^(n)\mathbb{A}^{n}An. For instance, the Russell cubic threefold, defined by the hypersurface equation x+x2y+z2+t3=0x+x2y+z2+t3=0x+x^(2)y+z^(2)+t^(3)=0x+x^{2} y+z^{2}+t^{3}=0x+x2y+z2+t3=0 is known to be nonisomorphic to affine space and also A1A1A^(1)\mathbb{A}^{1}A1-contractible [28]. See [16] for a survey of further examples.
Example 3.2. If f:X→Yf:X→Yf:X rarr Yf: X \rightarrow Yf:X→Y is a Nisnevich locally trivial morphism with fibers that are A1A1A^(1)\mathbb{A}^{1}A1-contractible smooth kkkkk-schemes, then fffff is an A1A1A^(1)\mathbb{A}^{1}A1-weak equivalence. Thus, the projection morphism for a vector bundle is an A1A1A^(1)\mathbb{A}^{1}A1-weak equivalence. A vector bundle EEEEE over a scheme XXXXX can be seen as a commutative algebraic XXXXX-group scheme, so we may speak of EEEEE-torsors; EEEEE-torsors are classified by the coherent cohomology group H1(X,E)H1(X,E)H^(1)(X,E)\mathrm{H}^{1}(X, \mathscr{E})H1(X,E) (in particular, vector bundle torsors over affine schemes may always be trivialized). Vector bundle torsors are Zariski locally trivial fiber bundles with fibers isomorphic to affine spaces, and the projection morphism for a vector bundle torsor is an A1A1A^(1)\mathbb{A}^{1}A1-weak equivalence.
By an affine vector bundle torsor over a scheme XXXXX we will mean a torsor π:Y→XÏ€:Y→Xpi:Y rarr X\pi: Y \rightarrow XÏ€:Y→X for some vector bundle EEEEE on XXXXX such that YYYYY is an affine scheme. Jouanolou proved [35, LEMME 1.5] that any quasiprojective variety admits an affine vector bundle torsor. Thomason [63, PROPOSITION 4.4] generalized Jouanolou's observation, and the following result is a special case of his results.
Lemma 3.3 (Jouanolou-Thomason homotopy lemma). If XXXXX is a smooth kkkkk-variety, then XXXXX admits an affine vector bundle torsor. In particular, any smooth kkkkk-variety is isomorphic in H(k)H(k)H(k)\mathrm{H}(k)H(k) to a smooth affine variety.
Definition 3.4. By a Jouanolou device for a smooth kkkkk-variety XXXXX we will mean a choice of an affine vector bundle torsor p:Y→Xp:Y→Xp:Y rarr Xp: Y \rightarrow Xp:Y→X.
Example 3.5. When X=PnX=PnX=P^(n)X=\mathbb{P}^{n}X=Pn there is a very simple construction of a "standard" Jouanolou device P~nP~ntilde(P)^(n)\tilde{\mathbb{P}}^{n}P~n. Geometrically, the standard Jouanolou device for PnPnP^(n)\mathbb{P}^{n}Pn may be described as the complement of the incidence divisor in Pn×PnPn×PnP^(n)xxP^(n)\mathbb{P}^{n} \times \mathbb{P}^{n}Pn×Pn where the second projective space is viewed as the dual of the first, with structure morphism the projection onto either factor.
Example 3.6. If XXXXX is a smooth projective variety of dimension ddddd, then we may choose a finite morphism ψ:X→Pdψ:X→Pdpsi:X rarrP^(d)\psi: X \rightarrow \mathbb{P}^{d}ψ:X→Pd. Pulling back the standard Jouanolou device for PdPdP^(d)\mathbb{P}^{d}Pd along ψψpsi\psiψ, we see that XXXXX admits a Jouanolou device X~X~tilde(X)\tilde{X}X~ of dimension 2d2d2d2 d2d.
Example 3.7. For n∈Nn∈Nn inNn \in \mathbb{N}n∈N, consider the smooth affine kkkkk-scheme Q2n−1Q2n−1Q_(2n-1)Q_{2 n-1}Q2n−1 defined as the hypersurface in Ak2nAk2nA_(k)^(2n)\mathbb{A}_{k}^{2 n}Ak2n given by the equation ∑i=1nxiyi=1∑i=1n xiyi=1sum_(i=1)^(n)x_(i)y_(i)=1\sum_{i=1}^{n} x_{i} y_{i}=1∑i=1nxiyi=1. Projecting onto the first nnnnn-factors, we obtain a map p:Q2n−1→An∖0p:Q2n−1→An∖0p:Q_(2n-1)rarrA^(n)\\0p: Q_{2 n-1} \rightarrow \mathbb{A}^{n} \backslash 0p:Q2n−1→An∖0 which one may check is an affine vector bundle torsor. For any integer n≥0,An∖0n≥0,An∖0n >= 0,A^(n)\\0n \geq 0, \mathbb{A}^{n} \backslash 0n≥0,An∖0 is A1A1A^(1)\mathbb{A}^{1}A1-weakly equivalent to Sn−1,nSn−1,nS^(n-1,n)S^{n-1, n}Sn−1,n (see [41, §3.2, EXAMPLE 2.20]) and consequently Q2n−1Q2n−1Q_(2n-1)Q_{2 n-1}Q2n−1 is A1A1A^(1)\mathbb{A}^{1}A1-weakly equivalent to Sn−1,nSn−1,nS^(n-1,n)S^{n-1, n}Sn−1,n as well.
Example 3.8. For n∈Nn∈Nn inNn \in \mathbb{N}n∈N, consider the smooth affine kkkkk-scheme Q2nQ2nQ_(2n)Q_{2 n}Q2n defined as the hypersurface in Ak2n+1Ak2n+1A_(k)^(2n+1)\mathbb{A}_{k}^{2 n+1}Ak2n+1 given by the equation
The variety Q2Q2Q_(2)Q_{2}Q2 is isomorphic to the standard Jouanolou device over P1P1P^(1)\mathbb{P}^{1}P1. The variety P1P1P^(1)\mathbb{P}^{1}P1 is A1A1A^(1)\mathbb{A}^{1}A1-weakly equivalent to S1,1S1,1S^(1,1)S^{1,1}S1,1 and therefore Q2Q2Q_(2)Q_{2}Q2 is A1A1A^(1)\mathbb{A}^{1}A1-weakly equivalent to S1,1S1,1S^(1,1)S^{1,1}S1,1 as well. For n≥2n≥2n >= 2n \geq 2n≥2, one knows that Q2nQ2nQ_(2n)Q_{2 n}Q2n is A1A1A^(1)\mathbb{A}^{1}A1-weakly equivalent to Sn,nSn,nS^(n,n)S^{n, n}Sn,n [2, THEOREM 2].
3.4. Representability results
If FFF\mathscr{F}F is a presheaf on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk, we will say that FFF\mathscr{F}F is A1A1A^(1)\mathbb{A}^{1}A1-invariant (resp. A1A1A^(1)\mathbb{A}^{1}A1-invariant on affines) if the pullback map F(X)→F(X×A1)F(X)→FX×A1F(X)rarrF(X xxA^(1))\mathscr{F}(X) \rightarrow \mathscr{F}\left(X \times \mathbb{A}^{1}\right)F(X)→F(X×A1) is an isomorphism for all X∈SmkX∈SmkX inSm_(k)X \in \operatorname{Sm}_{k}X∈Smk (resp. X∈Smkaff X∈Smkaff X inSm_(k)^("aff ")X \in \mathrm{Sm}_{k}^{\text {aff }}X∈Smkaff ). A necessary condition for a cohomology theory on smooth schemes to be representable in H(k)H(k)H(k)\mathrm{H}(k)H(k) is that it is A1A1A^(1)\mathbb{A}^{1}A1-invariant and has a Mayer-Vietoris property with respect to the Nisnevich topology. One of the first functors that one encounters with these properties is that which assigns to a smooth kkkkk-scheme its Picard group. Morel and Voevodsky showed [41, $4 PROPOSITION 3.8] that if XXXXX is a smooth kkkkk-scheme, then the A1A1A^(1)\mathbb{A}^{1}A1-weak equivalence P∞→BGmP∞→BGmP^(oo)rarrBG_(m)\mathbb{P}^{\infty} \rightarrow \mathrm{B} \mathbb{G}_{m}P∞→BGm induces a bijection [X,P∞]A1≅Pic(X)X,P∞A1≅Picâ¡(X)[X,P^(oo)]_(A^(1))~=Pic(X)\left[X, \mathbb{P}^{\infty}\right]_{\mathbb{A}^{1}} \cong \operatorname{Pic}(X)[X,P∞]A1≅Picâ¡(X).
If AAA\mathbf{A}A is a sheaf of abelian groups on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk, then the functors HNisi(−,A)HNisi(−,A)H_(Nis)^(i)(-,A)\mathrm{H}_{\mathrm{Nis}}^{i}(-, \mathbf{A})HNisi(−,A) frequently fail to be A1A1A^(1)\mathbb{A}^{1}A1-invariant (taking A=GaA=GaA=G_(a)\mathbf{A}=\mathbb{G}_{a}A=Ga gives a simple example) and therefore fail to be representable on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk. The situation above where A=GmA=GmA=G_(m)\mathbf{A}=\mathbb{G}_{m}A=Gm provides the prototypical example of a sheaf whose cohomology is A1A1A^(1)\mathbb{A}^{1}A1-invariant (here the zeroth cohomology is the presheaf of units, which is even A1A1A^(1)\mathbb{A}^{1}A1-invariant on reduced schemes). Following Morel and Voevodsky, we distinguish the cases where sheaf cohomology is A1A1A^(1)\mathbb{A}^{1}A1-invariant.
Definition 3.9. A sheaf of groups GGG\mathbf{G}G on SmkSmkSm_(k)\operatorname{Sm}_{k}Smk is called strongly A1A1A^(1)\mathbb{A}^{1}A1-invariant if for i=0,1i=0,1i=0,1i=0,1i=0,1 the functors HNisi(−,G)HNisi(−,G)H_(Nis)^(i)(-,G)\mathrm{H}_{\mathrm{Nis}}^{i}(-, \mathbf{G})HNisi(−,G) on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk are A1A1A^(1)\mathbb{A}^{1}A1-invariant. A sheaf of abelian groups AAA\mathbf{A}A on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk is called strictly A1A1A^(1)\mathbb{A}^{1}A1-invariant if for all i≥0i≥0i >= 0i \geq 0i≥0 the functors HNisi(−,A)HNisi(−,A)H_(Nis)^(i)(-,A)\mathrm{H}_{\mathrm{Nis}}^{i}(-, \mathbf{A})HNisi(−,A) on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk are A1A1A^(1)\mathbb{A}^{1}A1-invariant.
The fundamental work of Morel, which we will review shortly, demonstrates the key role played by strongly and strictly A1A1A^(1)\mathbb{A}^{1}A1-invariant sheaves. Nevertheless, various natural functors of geometric origin fail to be A1A1A^(1)\mathbb{A}^{1}A1-invariant on smooth schemes.
Example 3.10. If r≥2r≥2r >= 2r \geq 2r≥2, then the functor HNis1(−,GLr)HNis1−,GLrH_(Nis)^(1)(-,GL_(r))\mathrm{H}_{\mathrm{Nis}}^{1}\left(-, \mathrm{GL}_{r}\right)HNis1(−,GLr) fails to be A1A1A^(1)\mathbb{A}^{1}A1-invariant on all schemes. For an explicit example, consider the simplest case. By a theorem of DedekindWeber frequently attributed to Grothendieck every rank nnnnn vector bundle on P1P1P^(1)\mathbb{P}^{1}P1 is isomorphic to a unique line bundle of the form ⨁i=1nO(ai)â¨i=1n Oaibigoplus_(i=1)^(n)O(a_(i))\bigoplus_{i=1}^{n} \mathcal{O}\left(a_{i}\right)â¨i=1nO(ai) with the aiaia_(i)a_{i}ai weakly increasing. On the other hand, consider P1×A1P1×A1P^(1)xxA^(1)\mathbb{P}^{1} \times \mathbb{A}^{1}P1×A1 with coordinates ttttt and xxxxx. The matrix
(tx0t−1)tx0t−1([t,x],[0,t^(-1)])\left(\begin{array}{cc}
t & x \\
0 & t^{-1}
\end{array}\right)(tx0t−1)
determines a rank 2 vector bundle on P1×A1P1×A1P^(1)xxA^(1)\mathbb{P}^{1} \times \mathbb{A}^{1}P1×A1 whose restriction to P1×0P1×0P^(1)xx0\mathbb{P}^{1} \times 0P1×0 is O(1)⊕O(−1)O(1)⊕O(−1)O(1)o+O(-1)\mathcal{O}(1) \oplus \mathcal{O}(-1)O(1)⊕O(−1) and whose restriction to P1×1P1×1P^(1)xx1\mathbb{P}^{1} \times 1P1×1 is O⊕OO⊕OOo+O\mathcal{O} \oplus \mathcal{O}O⊕O. In contrast, Lindel's theorem affirming the BassQuillen conjecture in the geometric case shows that HNis1(−,GLr)HNis1−,GLrH_(Nis)^(1)(-,GL_(r))\mathrm{H}_{\mathrm{Nis}}^{1}\left(-, \mathrm{GL}_{r}\right)HNis1(−,GLr) is A1A1A^(1)\mathbb{A}^{1}A1-invariant on affines. The next result generalizes this last observation.
Theorem 3.11 (Morel, Schlichting, Asok-Hoyois-Wendt). If XXXXX is a smooth affine kkkkk-scheme, then for any r∈Nr∈Nr inNr \in \mathbb{N}r∈N there are functorial bijections of the form
Remark 3.12. The above result was first established by F. Morel in [40] for r≠2r≠2r!=2r \neq 2r≠2 and kkkkk an infinite, perfect field, and his proof was partly simplified by M. Schlichting whose argument also established the case r=2r=2r=2r=2r=2 [51]. The version above is stated in [13].
In [14,15], it is shown that if GGG\mathrm{G}G is an isotropic reductive group scheme (see [14, DEFINITION 3.3.5] for the definition), then the functor assigning to X∈Smkaff X∈Smkaff X inSm_(k)^("aff ")X \in \operatorname{Sm}_{k}^{\text {aff }}X∈Smkaff the set HNis1(X,G)HNis1(X,G)H_(Nis)^(1)(X,G)\mathrm{H}_{\mathrm{Nis}}^{1}(X, \mathrm{G})HNis1(X,G) is representable by BG. This observation has a number of consequences, e.g., the following result about quadrics (see Examples 3.7 and 3.8).
Theorem 3.14 ([1,14,15]). For any integer i≥1i≥1i >= 1i \geq 1i≥1 and any X∈Smkaff X∈Smkaff X inSm_(k)^("aff ")X \in \mathrm{Sm}_{k}^{\text {aff }}X∈Smkaff , the comparison map
is a bijection, contravariantly functorial in XXXXX.
3.5. Postnikov towers, connectedness and strictly A1A1A^(1)\mathbb{A}^{1}A1-invariant sheaves
Recall from Definition 3.9 the notion of strongly or strictly A1A1A^(1)\mathbb{A}^{1}A1-invariant sheaves of groups. F. Morel showed that such sheaves can be thought of as "building blocks" for the unstable A1A1A^(1)\mathbb{A}^{1}A1-homotopy category. Morel's foundational works [39,40][39,40][39,40][39,40][39,40] can be viewed as a careful analysis of strictly and strongly A1A1A^(1)\mathbb{A}^{1}A1-invariant sheaves of groups and the relationship between the two notions. More precisely, Morel showed that working over a perfect
field kkkkk, the A1A1A^(1)\mathbb{A}^{1}A1-homotopy sheaves of a motivic space are always strongly A1A1A^(1)\mathbb{A}^{1}A1-invariant, and that strongly A1A1A^(1)\mathbb{A}^{1}A1-invariant sheaves of abelian groups are automatically strictly A1A1A^(1)\mathbb{A}^{1}A1-invariant.
Example 3.15. Some examples of A1A1A^(1)\mathbb{A}^{1}A1-invariant sheaves that will appear in the sequel are:
unramified Milnor K-theory sheaves KiM,i≥0KiM,i≥0K_(i)^(M),i >= 0\mathbf{K}_{i}^{M}, i \geq 0KiM,i≥0 (see [50, coRoLLARY 6.5, PRoPOSItION 8.6] where, more generally, it is shown that any Rost cycle module gives rise to a strictly A1A1A^(1)\mathbb{A}^{1}A1-invariant sheaf);
the Witt sheaf WWW\mathbf{W}W or unramified powers of the fundamental ideal in the Witt ring IjIjI^(j)\mathbf{I}^{j}Ij, j≥0j≥0j >= 0j \geq 0j≥0 (this follows from [46]); and
unramified Milnor-Witt K-theory sheaves KiMW,i∈ZKiMW,i∈ZK_(i)^(MW),i inZ\mathbf{K}_{i}^{M W}, i \in \mathbb{Z}KiMW,i∈Z (see [40, cHAPTER 3] for this assertion, or [31, COROLLARY 8.5,PROPOSITION 9.1] where this observation is generalized to so-called Milnor-Witt cycle modules).
3.16 (Moore-Postnikov factorizations). There is an analog of the Moore-Postnikov factorization of a map f:E→Bf:E→Bf:ErarrBf: \mathscr{E} \rightarrow \mathscr{B}f:E→B of spaces along the lines described in Section 2. For concreteness we discuss the case where EEE\mathscr{E}E and BBB\mathscr{B}B are A1A1A^(1)\mathbb{A}^{1}A1-connected and fffff induces an isomorphism on A1A1A^(1)\mathbb{A}^{1}A1-fundamental sheaves of groups for some choice of base-point in EEE\mathscr{E}E.
Given fffff as above, there are τ≤if∈Spckτ≤if∈Spcktau_( <= i)f inSpc_(k)\tau_{\leq i} f \in \operatorname{Spc}_{k}τ≤if∈Spck together with maps E→τ≤if,τ≤if→BE→τ≤if,τ≤if→BErarrtau_( <= i)f,tau_( <= i)f rarrB\mathscr{E} \rightarrow \tau_{\leq i} f, \tau_{\leq i} f \rightarrow \mathscr{B}E→τ≤if,τ≤if→B and τ≤if→τ≤i−1fτ≤if→τ≤i−1ftau_( <= i)f rarrtau_( <= i-1)f\tau_{\leq i} f \rightarrow \tau_{\leq i-1} fτ≤if→τ≤i−1f fitting into a diagram of exactly the same form as (2.1) (replacing EEEEE by EEE\mathscr{E}E and BBBBB by B)B)B)\mathscr{B})B). The relevant properties of this presentation are similar to those sketched before (replacing homotopy groups by homotopy sheaves), together with a homotopy pullback diagram of exactly the same form as (2.2). We refer to this tower as the A1A1A^(1)\mathbb{A}^{1}A1-Moore-Postnikov tower of fffff and the reader may consult [40, APPENDIX B] or [5, §6] for a more detailed presentation.
If XXXXX is a smooth scheme, then a map ψ:X→Bψ:X→Bpsi:X rarrB\psi: X \rightarrow \mathscr{B}ψ:X→B lifts to ψ~:X→Eψ~:X→Etilde(psi):X rarrE\tilde{\psi}: X \rightarrow \mathscr{E}ψ~:X→E if and only if lifts exist at each stage of the tower, i.e., if and only if a suitable obstruction vanishes. These obstructions are, by construction, valued in Nisnevich cohomology on XXXXX with values in a strictly A1A1A^(1)\mathbb{A}^{1}A1-invariant sheaf (see [5,§6][5,§6][5,§6][5, \S 6]§[5,§6] for a more detailed explanation).
By analogy with the situation in topology, we will use the A1A1A^(1)\mathbb{A}^{1}A1-Moore-Postnikov factorization to study lifting problems by means of obstruction theory. The relevant obstructions will lie in cohomology groups of a smooth scheme with coefficients in a strictly A1A1A^(1)\mathbb{A}^{1}A1-invariant sheaf. This motivates the following definition.
Definition 3.17. Let XXXXX be a smooth kkkkk-scheme. We say that XXXXX has A1A1A^(1)\mathbb{A}^{1}A1-cohomological dimension ≤d≤d<= d\leq d≤d if for any integer i>di>di > di>di>d and any strictly A1A1A^(1)\mathbb{A}^{1}A1-invariant sheaf F,HNisi(X,F)=0F,HNisi(X,F)=0F,H_(Nis)^(i)(X,F)=0\mathbf{F}, \mathrm{H}_{\mathrm{Nis}}^{i}(X, \mathbf{F})=0F,HNisi(X,F)=0. In that case, we write cdA1(X)≤dcdA1(X)≤dcd_(A^(1))(X) <= dc d_{\mathbb{A}^{1}}(X) \leq dcdA1(X)≤d.
Example 3.18. If XXXXX is a smooth kkkkk-scheme of dimension ddddd, then XXXXX necessarily has A1A1A^(1)\mathbb{A}^{1}A1-cohomological dimension ≤d≤d<= d\leq d≤d as well. Since AnAnA^(n)\mathbb{A}^{n}An has A1A1A^(1)\mathbb{A}^{1}A1-cohomological dimension ≤0≤0<= 0\leq 0≤0, the A1A1A^(1)\mathbb{A}^{1}A1-cohomological dimension can be strictly smaller than Krull dimension; Example 3.6 gives numerous other such examples.
3.6. Complex realization
Assume kkkkk is a field that admits an embedding ιC:k↪CιC:k↪CiotaC:k↪C\iota \mathbb{C}: k \hookrightarrow \mathbb{C}ιC:k↪C. The functor that assigns to a smooth kkkkk-variety XXXXX the complex manifold X(C)X(C)X(C)X(\mathbb{C})X(C) equipped with its classical topology extends to a complex realization functor
where HHH\mathrm{H}H is the usual homotopy category of topological spaces [41, §3.3]. By construction, complex realization preserves finite products and homotopy colimits. It follows that the complex realization of the motivic sphere Sp,qSp,qS^(p,q)S^{p, q}Sp,q is the ordinary sphere Sp+qSp+qS^(p+q)S^{p+q}Sp+q, and consequently the complex realization functor induces group homomorphisms of the form
for any pointed smooth kkkkk-scheme (X,x)(X,x)(X,x)(X, x)(X,x).
Suppose XXXXX is any kkkkk-scheme admitting a complex embedding and fix such an embedding. Write Vectrtop (X)Vectrtop â¡(X)Vect_(r)^("top ")(X)\operatorname{Vect}_{r}^{\text {top }}(X)Vectrtop â¡(X) for the set of isomorphism classes of complex topological vector bundles on XXXXX. There is a function
sending an algebraic vector bundle EEEEE over XXXXX to the topological vector bundle on X(C)X(C)X(C)X(\mathbb{C})X(C) attached to the base change of EEEEE to XCXCX_(C)X_{\mathbb{C}}XC. We will say that an algebraic vector bundle is algebraizable if it lies in the image of this map.
As rank rrrrr topological vector bundles are classified by the set [X(C),BU(r)][X(C),BU(r)][X(C),BU(r)][X(\mathbb{C}), \mathrm{B} U(r)][X(C),BU(r)] of homotopy classes of maps from X(C)X(C)X(C)X(\mathbb{C})X(C) to the complex Grassmannian, it follows that the function of the preceding paragraph factors as
Theorem 3.11 implies that the first map is a bijection if XXXXX is a smooth affine kkkkk-scheme (or, alternatively, if r=1r=1r=1r=1r=1 ). More generally, combining Theorem 3.11 and Lemma 3.3 one knows that any element of [X,Grr]A1X,GrrA1[X,Gr_(r)]_(A^(1))\left[X, \mathrm{Gr}_{r}\right]_{\mathbb{A}^{1}}[X,Grr]A1 may be represented by an actual rank rrrrr vector bundle on any Jouanolou device X~X~tilde(X)\tilde{X}X~ of XXXXX; this suggests the following definition.
Definition 3.19. If XXXXX is a smooth kkkkk-scheme, then by a rank rrrrr motivic vector bundle on XXXXX we mean an element of the set [X,Grr]A1X,GrrA1[X,Gr_(r)]_(A^(1))\left[X, \mathrm{Gr}_{r}\right]_{\mathbb{A}^{1}}[X,Grr]A1.
Question 3.20. If XXXXX is a smooth complex algebraic variety, then which topological vector bundles are algebraizable (resp. motivic)?
4. OBSTRUCTION THEORY AND VECTOR BUNDLES
In order to apply the obstruction theory described in the previous sections to analyze algebraic vector bundles, we need more information about the structure of the classifying space BGLnBGLnBGL_(n)\mathrm{BGL}_{n}BGLn including information about its A1A1A^(1)\mathbb{A}^{1}A1-homotopy sheaves, and the structure of the homotopy fiber of the stabilization map BGLn→BGLn+1BGLn→BGLn+1BGL_(n)rarrBGL_(n+1)\mathrm{BGL}_{n} \rightarrow \mathrm{BGL}_{n+1}BGLn→BGLn+1 induced by the map GLn→GLn+1GLn→GLn+1GL_(n)rarrGL_(n+1)\mathrm{GL}_{n} \rightarrow \mathrm{GL}_{n+1}GLn→GLn+1 sending an invertible matrix XXXXX to the block matrix diag(1,X)diagâ¡(1,X)diag(1,X)\operatorname{diag}(1, X)diagâ¡(1,X).
4.1. The homotopy sheaves of the classifying space of BGLnBGLnBGL_(n)\mathrm{BGL}_{\boldsymbol{n}}BGLn
We observed earlier that BGL1=BGmBGL1=BGmBGL_(1)=BG_(m)\mathrm{BGL}_{1}=\mathrm{B} \mathbb{G}_{m}BGL1=BGm is an Eilenberg-Mac Lane space for the sheaf GmGmG_(m)\mathbb{G}_{m}Gm : it is A1A1A^(1)\mathbb{A}^{1}A1-connected, and has exactly 1 nonvanishing A1A1A^(1)\mathbb{A}^{1}A1-homotopy sheaf in degree 1 , which is isomorphic to GmGmG_(m)\mathbb{G}_{m}Gm. For n≥1n≥1n >= 1n \geq 1n≥1, the analysis of homotopy sheaves of BGLnBGLnBGL_(n)\mathrm{BGL}_{n}BGLn uses several ingredients. First, Morel-Voevodsky observed that BGL=colimnBGLnBGL=colimnâ¡BGLnBGL=colim_(n)BGL_(n)\mathrm{BGL}=\operatorname{colim}_{n} \mathrm{BGL}_{n}BGL=colimnâ¡BGLn (for the inclusions described above) represents (reduced) algebraic K-theory after [41, 84 THEOREM 3.13]. Second, Morel observed that there is an A1A1A^(1)\mathbb{A}^{1}A1-fiber sequence of the form
and that An+1∖0An+1∖0A^(n+1)\\0\mathbb{A}^{n+1} \backslash 0An+1∖0 is A1−(n−1)A1−(n−1)A^(1)-(n-1)\mathbb{A}^{1}-(n-1)A1−(n−1)-connected. Furthermore, Morel computed [40] the first nonvanishing A1A1A^(1)\mathbb{A}^{1}A1-homotopy sheaf of An+1∖0An+1∖0A^(n+1)\\0\mathbb{A}^{n+1} \backslash 0An+1∖0 in terms of what he called Milnor-Witt K-theory sheaves (Example 3.15).
Putting these ingredients together, one deduces
πiA1(BGLn)≅KiQ,1≤i≤n−1Ï€iA1BGLn≅KiQ,1≤i≤n−1pi_(i)^(A^(1))(BGL_(n))~=K_(i)^(Q),quad1 <= i <= n-1\pi_{i}^{\mathbb{A}^{1}}\left(\mathrm{BGL}_{n}\right) \cong \mathbf{K}_{i}^{Q}, \quad 1 \leq i \leq n-1Ï€iA1(BGLn)≅KiQ,1≤i≤n−1
where KiQKiQK_(i)^(Q)\mathbf{K}_{i}^{Q}KiQ is the (Nisnevich) sheafification of the Quillen K-theory presheaf on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk. Following terminology from topology, sheaves in this range are called stable, and the case i=ni=ni=ni=ni=n is called the first unstable homotopy sheaf. In [3], we described the first unstable homotopy sheaf of BGLnBGLnBGL_(n)\mathrm{BGL}_{n}BGLn.
The group scheme GLnGLnGL_(n)\mathrm{GL}_{n}GLn maps to GLn(C)GLn(C)GL_(n)(C)\mathrm{GL}_{n}(\mathbb{C})GLn(C) under complex realization; the latter is homotopy equivalent to U(n)U(n)U(n)U(n)U(n). For context, we recall some facts about homotopy of U(n)U(n)U(n)U(n)U(n). A classical result of Bott, refining results of Borel-Hirzeburch [24, THEOREM 25.8] asserts that the image of π2n(BU(n))Ï€2n(BU(n))pi_(2n)(BU(n))\pi_{2 n}(B U(n))Ï€2n(BU(n)) in H2n(BU(n))H2n(BU(n))H_(2n)(BU(n))H_{2 n}(B U(n))H2n(BU(n)) is divisible by precisely (n−1)(n−1)(n-1)(n-1)(n−1) ! [25]. This result implies the assertion that π2n(U(n))=nÏ€2n(U(n))=npi_(2n)(U(n))=n\pi_{2 n}(U(n))=nÏ€2n(U(n))=n !.
Complex realization yields a map πn,nA1(GLn)→π2n(U(n))Ï€n,nA1GLn→π2n(U(n))pi_(n,n)^(A^(1))(GL_(n))rarrpi_(2n)(U(n))\pi_{n, n}^{\mathbb{A}^{1}}\left(\mathrm{GL}_{n}\right) \rightarrow \pi_{2 n}(U(n))Ï€n,nA1(GLn)→π2n(U(n)). One can view the celebrated "Suslin matrices" [59] as providing an algebro-geometric realization of the generator of π2n(U(n))Ï€2n(U(n))pi_(2n)(U(n))\pi_{2 n}(U(n))Ï€2n(U(n)). Analyzing the fiber sequence of (4.1) and putting all of the ingredients above together, we obtain the following result (we refer the reader to Example 3.15 for notation).
Theorem 4.1 ([3, THEOREM 1.1]). Assume kkkkk is a field that has characteristic not equal to 2. For any integer n≥2n≥2n >= 2n \geq 2n≥2, there are strictly A1A1A^(1)\mathbb{A}^{1}A1-invariant sheaves SnSnS_(n)\mathbf{S}_{n}Sn fitting into exact sequences of the form:
0→Sn+1→πnA1(BGLn)→KnQ→0, oodd ;0→Sn+1×Kn+1M/2In+1→πnA1(BGLn)→KnQ→0, n even 0→Sn+1→πnA1BGLn→KnQ→0, oodd ;0→Sn+1×Kn+1M/2In+1→πnA1BGLn→KnQ→0, n even {:[0rarrS_(n+1),rarrpi_(n)^(A^(1))(BGL_(n)),rarrK_(n)^(Q)rarr0",",],[," oodd ";],[0rarrS_(n+1)xx_(K_(n+1)^(M)//2)I^(n+1),rarrpi_(n)^(A^(1))(BGL_(n)),rarrK_(n)^(Q)rarr0","," n even "]:}\begin{array}{rlrl}
0 \rightarrow \mathbf{S}_{n+1} & \rightarrow \pi_{n}^{\mathbb{A}^{1}}\left(\mathrm{BGL}_{n}\right) & \rightarrow \mathbf{K}_{n}^{Q} \rightarrow 0, & \\
& \text { oodd } ; \\
0 \rightarrow \mathbf{S}_{n+1} \times_{\mathbf{K}_{n+1}^{M} / 2} \mathbf{I}^{n+1} & \rightarrow \pi_{n}^{\mathbb{A}^{1}}\left(\mathrm{BGL}_{n}\right) & \rightarrow \mathbf{K}_{n}^{Q} \rightarrow 0, & \text { n even }
\end{array}0→Sn+1→πnA1(BGLn)→KnQ→0, oodd ;0→Sn+1×Kn+1M/2In+1→πnA1(BGLn)→KnQ→0, n evenÂ
where
(1) there is a canonical epimorphism KnM/(n−1)!→SnKnM/(n−1)!→SnK_(n)^(M)//(n-1)!rarrS_(n)\mathbf{K}_{n}^{M} /(n-1)!\rightarrow \mathbf{S}_{n}KnM/(n−1)!→Sn which becomes an isomorphism after n−2n−2n-2n-2n−2 contractions (see [3,§2.3][3,§2.3][3,§2.3][3, \S \mathbf{2} . \mathbf{3}]§[3,§2.3] for this terminology);
(2) there is a canonical epimorphism Sn→KnM/2Sn→KnM/2S_(n)rarrK_(n)^(M)//2\mathbf{S}_{n} \rightarrow \mathbf{K}_{n}^{M} / 2Sn→KnM/2 such that the composite
(3) the fiber product is taken over the epimorphism Sn+1→Kn+1M/2Sn+1→Kn+1M/2S_(n+1)rarrK_(n+1)^(M)//2\mathbf{S}_{n+1} \rightarrow \mathbf{K}_{n+1}^{M} / 2Sn+1→Kn+1M/2 and a sheafified version of Milnor's homomorphism In+1→Kn+1M/2In+1→Kn+1M/2I^(n+1)rarrK_(n+1)^(M)//2\mathbf{I}^{n+1} \rightarrow \mathbf{K}_{n+1}^{M} / 2In+1→Kn+1M/2.
Moreover, if kkkkk admits a complex embedding, then the map
Bott's refinement of the theorem of Borel-Hirzebruch turns out to have an algebrogeometric interpretation. Indeed, in joint work with T. B. Williams [12] we showed that SnSnS_(n)\mathbf{S}_{n}Sn can described using a "Hurewicz map" analyzed by Andrei Suslin [60]. Suslin's conjecture on the image of this map is equivalent to the following conjecture.
Conjecture 4.2 (Suslin's factorial conjecture). The canonical epimorphism KnMKnMK_(n)^(M)\mathbf{K}_{n}^{M}KnM / (n−1)!→Sn(n−1)!→Sn(n-1)!rarrS_(n)(n-1)!\rightarrow \mathbf{S}_{n}(n−1)!→Sn is an isomorphism.
Remark 4.3. The conjecture holds tautologically for n=2n=2n=2n=2n=2. For n=3n=3n=3n=3n=3, Suslin observed the conjecture was equivalent to the Milnor conjecture on quadratic forms, which was resolved later independently by Merkurjev-Suslin and Rost. The conjecture was established for n=5n=5n=5n=5n=5 in "most" cases in [12] (see the latter for a precise statement); this work relies heavily on the computation by Østvær-Röndigs-Spitzweck of the motivic stable 1-stem [49].
4.2. Splitting bundles, Euler classes, and cohomotopy
Morel's computations around An∖0An∖0A^(n)\\0\mathbb{A}^{n} \backslash 0An∖0 in conjunction with the fiber sequences of (4.1) allow a significant improvement of Serre's celebrated splitting theorem for smooth affine varieties over a field that we stated in the introduction.
Proof of the motivic Serre Splitting Theorem 1.1. Suppose XXXXX is a smooth affine kkkkk-variety having A1A1A^(1)\mathbb{A}^{1}A1-cohomological dimension ≤d≤d<= d\leq d≤d, and suppose ξ:X→BGLrξ:X→BGLrxi:X rarrBGL_(r)\xi: X \rightarrow \mathrm{BGL}_{r}ξ:X→BGLr classifies a rank r>dr>dr > dr>dr>d vector bundle on XXXXX. We proceed by analyzing the A1A1A^(1)\mathbb{A}^{1}A1-Moore-Postnikov factorization of the stabilization map (4.1) with n=r−1n=r−1n=r-1n=r-1n=r−1. In that case, combining the fact that Ar∖0Ar∖0A^(r)\\0\mathbb{A}^{r} \backslash 0Ar∖0 is A1A1A^(1)\mathbb{A}^{1}A1 - (r−2)(r−2)(r-2)(r-2)(r−2)-connected and the A1A1A^(1)\mathbb{A}^{1}A1-cohomological dimension assumption on XXXXX, one sees all obstructions to splitting vanish.
Remark 4.4. The proof of this result does not rely on the Serre splitting theorem. Since A1A1A^(1)\mathbb{A}^{1}A1-cohomological dimension can be strictly smaller than Krull dimension (Example 3.18), this statement is strictly stronger than Serre splitting. Importantly, the improvement achieved here seems inaccessible to classical techniques.
The algebro-geometric splitting problem in corank 0 on smooth affine varieties of dimension ddddd over a field kkkkk has been analyzed by many authors. When kkkkk is an algebraically closed field, M. P. Murthy proved that the top Chern class in Chow groups is the only obstruction to splitting [42]. When kkkkk is not algebraically closed, vanishing of the top Chern class is known to be insufficient to guarantee splitting, and Nori proposed some ideas to analyze this situation. His ideas led Bhatwadekar and Sridharan [23] to introduce what they called Euler class groups and to provide one explicit "generators and relations" answer to this question. At the same time, F. Morel proposed an approach to the splitting problem in corank 0 , which we recall here.
Theorem 4.5 (Morel's splitting theorem [40, THEOREM 1.32]). Assume kkkkk is a field and XXXXX is a smooth affine kkkkk-variety of A1A1A^(1)\mathbb{A}^{1}A1-cohomological dimension ≤d≤d<= d\leq d≤d. If EEE\mathcal{E}E is a rank ddddd vector bundle on XXXXX, then EEE\mathcal{E}E splits off a free rank 1 summand if and only if an Euler class e(E)∈HNisd(X,KdMW(detE))e(E)∈HNisdX,KdMW(detâ¡E)e(E)inH_(Nis)^(d)(X,K_(d)^(MW)(det E))e(\mathcal{E}) \in \mathrm{H}_{\mathrm{Nis}}^{d}\left(X, \mathbf{K}_{d}^{M W}(\operatorname{det} \mathcal{E})\right)e(E)∈HNisd(X,KdMW(detâ¡E)) vanishes.
Remark 4.6. The Euler class of Theorem 4.5 is precisely the first nonvanishing obstruction class, as described in Paragraph 3.16. A related "cohomological" approach to the splitting problem in corank 0 was proposed by Barge-Morel [20] and analyzed in the thesis of the second author [29]. The cohomological approach was in most cases shown to be equivalent to the "obstruction-theoretic" approach in [6]. We also refer the reader to [51] for related results on the theory of Euler classes, extending also to singular varieties.
The next result shows that the relationship between Euler classes à la BhatwadekarSridharan and Euler classes à la Morel is mediated by another topologically inspired notion: cohomotopy (at least for bundles of trivial determinant).
Theorem 4.7 ([8, Ñ‚heorem 1]). Suppose kkkkk is a field, nnnnn and ddddd are integers, n≥2n≥2n >= 2n \geq 2n≥2, and XXXXX is a smooth affine kkkkk-scheme of dimension d≤2n−2d≤2n−2d <= 2n-2d \leq 2 n-2d≤2n−2. Write En(X)En(X)E^(n)(X)\mathrm{E}^{n}(X)En(X) for the BhatwadekarSridharan Euler class group.
The set [X,Q2n]A1X,Q2nA1[X,Q_(2n)]_(A^(1))\left[X, Q_{2 n}\right]_{\mathbb{A}^{1}}[X,Q2n]A1 carries a functorial abelian group structure;
where the "Segre class" homomorphism s is surjective and an isomorphism if kkkkk is infinite and d≥2d≥2d >= 2d \geq 2d≥2, and the Hurewicz homomorphism hhhhh is an isomorphism if d≤nd≤nd <= nd \leq nd≤n.
Remark 4.8. The group structure on [X,Q2n]A1X,Q2nA1[X,Q_(2n)]_(A^(1))\left[X, Q_{2 n}\right]_{\mathbb{A}^{1}}[X,Q2n]A1 is an algebro-geometric variant of Borsuk's group structure on cohomotopy. The second point of the statement includes the algebrogeometric analog of the Hopf classification theorem from topology.
4.3. The next nontrivial A1A1A^(1)\mathbb{A}^{1}A1-homotopy sheaf of spheres
In Section 2.2 we described a cohomological approach to the splitting problem in corank 1 for smooth closed manifolds of dimension ddddd; this approach relied on the computa-
tion of πd(Sd−1)Ï€dSd−1pi_(d)(S^(d-1))\pi_{d}\left(S^{d-1}\right)Ï€d(Sd−1). In order to analyze the algebro-geometric splitting problem in corank 1 using the A1A1A^(1)\mathbb{A}^{1}A1-Moore-Postnikov factorization we will need as input further information about the homotopy sheaves of Ad∖0Ad∖0A^(d)\\0\mathbb{A}^{d} \backslash 0Ad∖0. We now describe known results in this direction. For technical reasons, we assume 2 is invertible in what follows.
4.3.1. The KOKOKOK OKO-degree map
In classical algebraic topology, all of the "low degree" elements in the homotopy of spheres can be realized by constructions of "linear algebraic" nature. The situation in algebraic geometry appears to be broadly similar. The first contribution to the "next" nontrivial homotopy sheaves of motivic spheres requires recalling the geometric formulation of Bott periodicity for Hermitian K-theory given by Schlichting-Tripathi.
This homomorphism is trivial if i<n−1i<n−1i < n-1i<n-1i<n−1 by connectivity estimates. If i=n−1i=n−1i=n-1i=n-1i=n−1, via Morel's calculations one obtains a morphism KnMW→GWnnKnMW→GWnnK_(n)^(MW)rarrGW_(n)^(n)\mathbf{K}_{n}^{\mathrm{MW}} \rightarrow \mathbf{G W}_{n}^{n}KnMW→GWnn whose sections over finitely generated field extensions of kkkkk can be viewed as a quadratic enhancement of the "natural" map from Milnor KKKKK-theory to Quillen KKKKK-theory defined by symbols; we will refer to it as the natural homomorphism (the natural homomorphism is known to be an isomorphism if n≤4n≤4n <= 4n \leq 4n≤4; the case n≤2n≤2n <= 2n \leq 2n≤2 is essentially Suslin's, n=3n=3n=3n=3n=3 is [7, THEOREM 4.3.1], and n=4n=4n=4n=4n=4 is unpublished work of O. Röndigs).
where the right-hand term is by definition a higher Grothendieck-Witt sheaf (obtained by sheafifying the corresponding higher Grothendieck-Witt presheaf on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk ). The above map
is an epimorphism for n=2,3n=2,3n=2,3n=2,3n=2,3 and it follows from these observations that the morphism is an epimorphism after (n−3)(n−3)(n-3)(n-3)(n−3) contractions [7, THEOREM 4.4.5].
4.3.2. The motivic J-homomorphism
The classical J-homomorphism has an algebro-geometric counterpart that yields the second contribution to the "next" homotopy sheaf of motivic spheres. The standard action of SLnSLnSL_(n)\mathrm{SL}_{n}SLn on AnAnA^(n)\mathbb{A}^{n}An extends to an action on the one-point compactification Pn/Pn−1Pn/Pn−1P^(n)//P^(n-1)\mathbb{P}^{n} / \mathbb{P}^{n-1}Pn/Pn−1. The latter space is a motivic sphere P1∧nP1∧nP^(1^(^^n))\mathbb{P}^{1^{\wedge n}}P1∧n and thus one obtains a map
ΣP1nSLn→P1∧nΣP1nSLn→P1∧nSigma_(P^(1))^(n)SL_(n)rarrP^(1^(^^n))\Sigma_{\mathbb{P}^{1}}^{n} S L_{n} \rightarrow \mathbb{P}^{1^{\wedge n}}ΣP1nSLn→P1∧n
As SLnSLnSL_(n)\mathrm{SL}_{n}SLn is A1A1A^(1)\mathbb{A}^{1}A1-connected, it follows that ΣP1nSLnΣP1nSLnSigma_(P^(1))^(n)SL_(n)\Sigma_{\mathbb{P}^{1}}^{n} \mathrm{SL}_{n}ΣP1nSLn is A1A1A^(1)\mathbb{A}^{1}A1 - nnnnn-connected.
The first nonvanishing A1A1A^(1)\mathbb{A}^{1}A1-homotopy sheaf appears in degree n+1n+1n+1n+1n+1; for n=2n=2n=2n=2n=2, it is isomorphic to K4MWK4MWK_(4)^(MW)\mathbf{K}_{4}^{M W}K4MW, while for n≥3n≥3n >= 3n \geq 3n≥3 it is isomorphic to Kn+2MKn+2MK_(n+2)^(M)\mathbf{K}_{n+2}^{M}Kn+2M; this follows from A1A1A^(1)\mathbb{A}^{1}A1-Hurewicz theorem combined with [17, PROPOSItIoN 3.3.9] using the fact that π1A1(SLn)=K2MÏ€1A1SLn=K2Mpi_(1)^(A^(1))(SL_(n))=K_(2)^(M)\pi_{1}^{\mathbb{A}^{1}}\left(\mathrm{SL}_{n}\right)=\mathbf{K}_{2}^{M}Ï€1A1(SLn)=K2M for n≥3n≥3n >= 3n \geq 3n≥3 and properties of the A1A1A^(1)\mathbb{A}^{1}A1-tensor product [17, LEMMA 5.1.8].
Combining the above discussion with that of the previous section, we see that for n≥3n≥3n >= 3n \geq 3n≥3, we may consider the composite maps Kn+2M→πn+1A1(P1∧n)→GWn+1nKn+2M→πn+1A1P1∧n→GWn+1nK_(n+2)^(M)rarrpi_(n+1)^(A^(1))(P^(1^^n))rarrGW_(n+1)^(n)\mathbf{K}_{n+2}^{M} \rightarrow \pi_{n+1}^{\mathbb{A}^{1}}\left(\mathbb{P}^{1 \wedge n}\right) \rightarrow \mathbf{G W}_{n+1}^{n}Kn+2M→πn+1A1(P1∧n)→GWn+1n; this composite is known to be zero, but the map induced by the J-homomorphism fails to be injective. Instead, it factors through a morphism
Furthermore, the map on the right fails to be surjective. The unstable description above is not present in the literature, but it is equivalent to the results stated in [12]. In [49], the stable motivic 1-stem was computed in the terms above: the above sequence is exact on the left stably. The next result compares the unstable group to the corresponding stable group.
Theorem 4.9. For any integer n≥3n≥3n >= 3n \geq 3n≥3, the kernel Un+1Un+1U_(n+1)\mathbf{U}_{n+1}Un+1 of the stabilization map
is a direct summand; the stabilization map is an isomorphism if n=3n=3n=3n=3n=3, i.e., U4=0U4=0U_(4)=0\mathbf{U}_{4}=0U4=0.
Conjecture 4.10. For n≥4n≥4n >= 4n \geq 4n≥4, the sheaf Un+1Un+1U_(n+1)\mathbf{U}_{n+1}Un+1 is zero.
Remark 4.11. Conjecture 4.10 would follow from a suitable version of the Freudenthal suspension theorem for P1P1P^(1)\mathbb{P}^{1}P1-suspension.
4.4. Splitting in corank 1
Using the results above, we can analyze the splitting problem for vector bundles in corank 1. The expected result was posed as a question by Murthy [43, P. 173] which we stated in the introduction as Conjecture 1.2. Murthy's conjecture is trivial if d=2d=2d=2d=2d=2. In [4] and [5] we established the following result, which reduces Murthy's question to Conjecture 4.10.
Theorem 4.12. Let XXXXX be a smooth affine scheme of dimension d≥2d≥2d >= 2d \geq 2d≥2 over an algebraically closed field kkkkk. A rank d−1d−1d-1d-1d−1 vector bundle &&&\&& on XXXXX splits off a trivial rank 1 summand if and
only if cd−1(E)∈CHd−1(X)cd−1(E)∈CHd−1(X)c_(d-1)(E)inCH^(d-1)(X)c_{d-1}(\mathcal{E}) \in \mathrm{CH}^{d-1}(X)cd−1(E)∈CHd−1(X) is trivial and a secondary obstruction
vanishes. This secondary obstruction vanishes if d=3,4d=3,4d=3,4d=3,4d=3,4 or if Conjecture 4.10 has a positive answer.
To establish this result, one uses the assumptions that XXXXX is smooth affine of Krull dimension ddddd and kkkkk is algebraically closed in a strong way. Indeed, these assertions can be leveraged to show that the primary obstruction, which is a priori an Euler class, actually coincides with the (d−1)(d−1)(d-1)(d-1)(d−1) st Chern class. The secondary obstruction can be described by Theorem 4.9 and the form of the secondary obstruction is extremely similar to Liao's description in Section 2.2: it is a coset in Chd(X)/(Sq2+c1(E)∪)Chd−1(X)Chd(X)/Sq2+c1(E)∪Chd−1(X)Ch^(d)(X)//(Sq^(2)+c_(1)(E)uu)Ch^(d-1)(X)\mathrm{Ch}^{d}(X) /\left(\mathrm{Sq}^{2}+c_{1}(\mathcal{E}) \cup\right) \mathrm{Ch}^{d-1}(X)Chd(X)/(Sq2+c1(E)∪)Chd−1(X) where Chi(X)=CHi(X)/2Chi(X)=CHi(X)/2Ch^(i)(X)=CH^(i)(X)//2\mathrm{Ch}^{i}(X)=\mathrm{CH}^{i}(X) / 2Chi(X)=CHi(X)/2. Once more, the assumptions on XXXXX guarantee that Chd(X)Chd(X)Ch^(d)(X)\mathrm{Ch}^{d}(X)Chd(X) is trivial and thus the secondary obstruction is so as well.
4.5. The enumeration problem
If a vector bundle EEEEE splits off a free rank 1 summand, then another natural question is to enumerate the possible E′E′E^(')E^{\prime}E′ that become isomorphic to EEEEE after adding a free rank 1 summand. This problem may also be analyzed in homotopy theoretic terms as it amounts to enumerating the number of distinct lifts. This kind of problem was studied in detail in topology by James and Thomas [33] and the same kind of analysis can be pursued in algebraic geometry.
The history of the enumeration problem in algebraic geometry goes back to early days of algebraic K-theory. Indeed, the Bass-Schanuel cancellation theorem [22] solves the enumeration problem for bundles of negative corank. Suslin's celebrated cancellation theorem [58] solved the enumeration problem in corank 0 . In all of these statements, "cancellation" means that there is a unique lift. On the other hand, Mohan Kumar observed [38] that for bundles of corank 2 , uniqueness was no longer true in general. Nevertheless, Suslin conjectured that the enumeration problem had a particularly nice solution in corank 1.
Conjecture 4.13 (Suslin's cancellation conjecture). If kkkkk is an algebraically closed field, and XXXXX is a smooth affine kkkkk-scheme of dimension d≥2d≥2d >= 2d \geq 2d≥2. If EEE\mathcal{E}E and E′E′E^(')\mathcal{E}^{\prime}E′ are corank 1 bundles that become isomorphic after addition of a trivial rank 1 summand, then EEE\mathcal{E}E and E′E′E^(')\mathcal{E}^{\prime}E′ are isomorphic.
The above conjecture is trivial when d=2d=2d=2d=2d=2. It was established for EEE\mathcal{E}E the trivial bundle of rank d−1d−1d-1d-1d−1 in [30] and d=dim(X)d=dimâ¡(X)d=dim(X)d=\operatorname{dim}(X)d=dimâ¡(X) under the condition that (d−1)(d−1)(d-1)(d-1)(d−1) ! is invertible in kkkkk. The above conjecture was also established for d=3d=3d=3d=3d=3 in [4] (assuming 2 is invertible in kkkkk ). Paralleling the results of James-Thomas in topology [33], P. Du was able to prove in [27] that Suslin's question has a positive answer for oriented vector bundles in case the cohomology group HNisd(X,πdA1(Ad∖0))HNisdX,Ï€dA1Ad∖0H_(Nis)^(d)(X,pi_(d)^(A^(1))(A^(d)\\0))\mathrm{H}_{\mathrm{Nis}}^{d}\left(X, \pi_{d}^{\mathbb{A}^{1}}\left(\mathbb{A}^{d} \backslash 0\right)\right)HNisd(X,Ï€dA1(Ad∖0)) vanishes. This vanishing statement would follow immediately from Conjecture 4.10.
5. VECTOR BUNDLES: NONAFFINE VARIETIES AND ALGEBRAIZABILITY
In this final section, we survey some joint work with M. J. Hopkins related to the classification of motivic vector bundles (see Definition 3.19), its relationship to the algebraizability question (see Question 3.20), and investigate the extent to which A1A1A^(1)\mathbb{A}^{1}A1-homotopy theory can be used to analyze vector bundles on projective varieties.
5.1. Descent along a Jouanolou device
If XXXXX is a smooth algebraic kkkkk-variety, then there is always the map
from rank rrrrr vector bundles to rank rrrrr motivic vector bundles. When XXXXX is affine, Theorem 3.11 guarantees that this map is a bijection, and examples show that the map fails to be an isomorphism outside of this case. Nevertheless, it is very interesting to try to quantify the failure of the above map to be a bijection.
If π:X~→XÏ€:X~→Xpi: tilde(X)rarr X\pi: \tilde{X} \rightarrow XÏ€:X~→X is a Jouanolou device for XXXXX, then it follows from the definitions that the map (5.1) coincides with π∗:Vectr(X)→Vectr(X~)π∗:Vectrâ¡(X)→Vectrâ¡(X~)pi^(**):Vect_(r)(X)rarrVect_(r)( tilde(X))\pi^{*}: \operatorname{Vect}_{r}(X) \rightarrow \operatorname{Vect}_{r}(\tilde{X})π∗:Vectrâ¡(X)→Vectrâ¡(X~) under the bijection of Theorem 3.11. The morphism πÏ€pi\piÏ€ is faithfully flat by construction, and therefore, vector bundles on XXXXX are precisely vector bundles on X~X~tilde(X)\tilde{X}X~ equipped with a descent datum along πÏ€pi\piÏ€.
Since π:X~→XÏ€:X~→Xpi: tilde(X)rarr X\pi: \tilde{X} \rightarrow XÏ€:X~→X is an affine morphism, it follows that X~×XX~X~×XX~tilde(X)xx_(X) tilde(X)\tilde{X} \times_{X} \tilde{X}X~×XX~ is itself an affine scheme, and the two projections p1,p2:X~×XX~→X~p1,p2:X~×XX~→X~p_(1),p_(2): tilde(X)xx_(X) tilde(X)rarr tilde(X)p_{1}, p_{2}: \tilde{X} \times_{X} \tilde{X} \rightarrow \tilde{X}p1,p2:X~×XX~→X~ are A1A1A^(1)\mathbb{A}^{1}A1-weak equivalences. Thus, pullbacks p1∗p1∗p_(1)^(**)p_{1}^{*}p1∗ and p2∗p2∗p_(2)^(**)p_{2}^{*}p2∗ are bijections on sets of isomorphism classes of vector bundles. In fact, since the relative diagonal map splits the two projections, the two pullbacks actually coincide on isomorphism classes. In descent-theoretic terms, these observations mean that any vector bundle EEE\mathscr{E}E on X~X~tilde(X)\tilde{X}X~ can always be equipped with an isomorphism p1∗E→∼p2∗Ep1∗E→∼p2∗Ep_(1)^(**)Erarr"∼"p_(2)^(**)Ep_{1}^{*} \mathscr{E} \xrightarrow{\sim} p_{2}^{*} \mathscr{E}p1∗E→∼p2∗E, i.e., a predescent datum. Thus, the only obstruction to descending a vector bundle along πÏ€pi\piÏ€ is whether one may choose a predescent datum that actually satisfies the cocycle condition. With this observation in mind, it seems natural to analyze the question of whether every vector bundle can be equipped with a descent datum along πÏ€pi\piÏ€.
Question 5.1. If XXXXX is a smooth kkkkk-variety and π:X~→XÏ€:X~→Xpi: tilde(X)rarr X\pi: \tilde{X} \rightarrow XÏ€:X~→X is a Jouanolou device for XXXXX, then is the pull-back map
Theorem 5.2 (Asok, Fasel, Hopkins). Suppose XXXXX is a smooth projective kkkkk-variety of dimension d. If either (i) d≤2d≤2d <= 2d \leq 2d≤2 or (ii) kkkkk is algebraically closed and d≤3d≤3d <= 3d \leq 3d≤3, then Question 5.1 admits a positive answer, i.e., every vector bundle on X~X~tilde(X)\tilde{X}X~ admits a descent datum relative to πÏ€pi\piÏ€.
5.2. Algebraizability I: obstructions
If XXXXX is a smooth complex algebraic variety, then we considered the map
and posed the question of characterizing its image. We observed that this map factors through the set of motivic vector bundles, so one necessary condition for a topological vector bundle to be algebraizable is that it admits a motivic lift. In particular, this means that the Chern classes of the topological vector bundle in integral cohomology must lie in the image of the cycle class map. It is natural to ask if algebraizability of Chern classes is sufficient to guarantee that a vector bundle admits a motivic left.
In case XXXXX is projective, this question has been for instance studied in [53] where it is proved that any vector bundle with algebraic Chern classes is algebraizable if dim(X)=2dimâ¡(X)=2dim(X)=2\operatorname{dim}(X)=2dimâ¡(X)=2. In case of projective threefolds, positive results are given by Atiyah-Rees and Bănică-Putinar respectively in [18] and [19]. If XXXXX is affine, the works of Swan-Murthy [44] and MurthyKumar [36] show that the answer to the question is positive if XXXXX is of dimension ≤3≤3<= 3\leq 3≤3 as a consequence of the following statement: Given any pair (α1,α2)∈CH1(X)×CH2(X)α1,α2∈CH1(X)×CH2(X)(alpha_(1),alpha_(2))inCH^(1)(X)xxCH^(2)(X)\left(\alpha_{1}, \alpha_{2}\right) \in \mathrm{CH}^{1}(X) \times \mathrm{CH}^{2}(X)(α1,α2)∈CH1(X)×CH2(X), there exists a vector bundle EEE\mathcal{E}E on XXXXX with ci(E)=αici(E)=αic_(i)(E)=alpha_(i)c_{i}(\mathscr{E})=\alpha_{i}ci(E)=αi. However, in dimension 4 , additional restrictions on Chern classes arise from the action of the motivic Steenrod algebra.
Theorem 5.3 ([10, THEOREM 2]). If XXXXX is a smooth affine 4-fold, then a pair (c1,c2)∈c1,c2∈(c_(1),c_(2))in\left(c_{1}, c_{2}\right) \in(c1,c2)∈CH1(X)×CH2(X)CH1(X)×CH2(X)CH^(1)(X)xxCH^(2)(X)\mathrm{CH}^{1}(X) \times \mathrm{CH}^{2}(X)CH1(X)×CH2(X) are Chern classes of a rank 2 bundle on XXXXX if and only if c1,c2c1,c2c_(1),c_(2)c_{1}, c_{2}c1,c2 satisfy the additional condition Sq2(c2)+c1c2=0Sq2c2+c1c2=0Sq^(2)(c_(2))+c_(1)c_(2)=0\mathrm{Sq}^{2}\left(c_{2}\right)+c_{1} c_{2}=0Sq2(c2)+c1c2=0, where
is the Steenrod squaring operation, and c1c2c1c2c_(1)c_(2)c_{1} c_{2}c1c2 is the reduction modulo 2 of the cup product.
Remark 5.4. This obstruction is sufficient to identify topological vector bundles on a smooth affine fourfold XXXXX having algebraic Chern classes which are not algebraizable [10, coRolLARY 3.1.5]. One example of such an XXXXX is provided by the open complement in P1×P3P1×P3P^(1)xxP^(3)\mathbb{P}^{1} \times \mathbb{P}^{3}P1×P3 of a suitable smooth hypersurface ZZZZZ of bidegree (3, 4).
5.3. Algebraizability II: building motivic vector bundles
The notion of a cellular space goes back to the work of Dror Farjoun. By a cellular motivic space, we will mean a space that can be built out of the motivic spheres Sp,qSp,qS^(p,q)S^{p, q}Sp,q by formation of homotopy colimits. It is straightforward to see inductively that PnPnP^(n)\mathbb{P}^{n}Pn is cellular. In the presence of cellularity assumptions, many obstructions to producing a motivic lift of a vector bundle vanish and this motivates the following conjecture.
Conjecture 5.5. If XXXXX is a smooth cellular CCC\mathbb{C}C-variety, then the map
Remark 5.6. The conjecture holds for PnPnP^(n)\mathbb{P}^{n}Pn for n≤3n≤3n <= 3n \leq 3n≤3 (this follows, for example, from the results of Schwarzenberger and Atiyah-Rees mentioned above); in these cases, bijectivity holds. For P4P4P^(4)\mathbb{P}^{4}P4, the "surjective" formulation of Conjecture 5.5 is known, but the "bijective" formulation is not.
We now analyze Conjecture 5.5 for a class of "interesting" topological vector bundles on PnPnP^(n)\mathbb{P}^{n}Pn introduced by E. Rees and L. Smith. We briefly recall the construction of these topological vector bundles here. By a classical result of Serre [54, PROPOsition 11], we know that if ppppp is a prime, then the ppppp-primary component of π4p−3(S3)Ï€4p−3S3pi_(4p-3)(S^(3))\pi_{4 p-3}\left(S^{3}\right)Ï€4p−3(S3) is isomorphic to Z/pZ/pZ//p\mathbb{Z} / pZ/p, generated by the composite of a generator α1α1alpha_(1)\alpha_{1}α1 of the ppppp-primary component of π2p(S3)Ï€2pS3pi_(2p)(S^(3))\pi_{2 p}\left(S^{3}\right)Ï€2p(S3) and the (2p−3)(2p−3)(2p-3)(2 p-3)(2p−3) rd suspension of itself; we will write α12α12alpha_(1)^(2)\alpha_{1}^{2}α12 for this class.
The map Pn→S2nPn→S2nP^(n)rarrS^(2n)\mathbb{P}^{n} \rightarrow S^{2 n}Pn→S2n that collapses Pn−1Pn−1P^(n-1)\mathbb{P}^{n-1}Pn−1 to a point determines a function
Rees established that the class α12α12alpha_(1)^(2)\alpha_{1}^{2}α12 determines a nontrivial rank 2 vector bundle ξp∈ξp∈xi_(p)in\xi_{p} \inξp∈[P2p−1,BSU(2)]P2p−1,BSU(2)[P^(2p-1),BSU(2)]\left[\mathbb{P}^{2 p-1}, \mathrm{BSU}(2)\right][P2p−1,BSU(2)]; we will refer to this bundle as a Rees bundle [48]. By construction, ξpξpxi_(p)\xi_{p}ξp is a nontrivial rank 2 bundle with trivial Chern classes.
The motivation for Rees' construction originated from results of Grauert-Schneider [32]. If the bundles ξpξpxi_(p)\xi_{p}ξp were algebraizable, then the fact that they have trivial Chern classes would imply they were necessarily unstable by Barth's results on Chern classes of stable vector bundles [21, CORoLLARY 1 P. 127] (here, stability means slope stability in the sense of Mumford). Grauert and Schneider analyzed unstable rank 2 vector bundles on projective space and they aimed to prove that such vector bundles were necessarily direct sums of line bundles; this assertion is now sometimes known as the Grauert-Schneider conjecture. In view of the Grauert-Schneider conjecture, the bundles ξpξpxi_(p)\xi_{p}ξp should not be algebraizable. On the other hand, one of the motivations for Conjecture 5.5 is the following result.
Theorem 5.7 ([11, THEOREM 2.2.16]). For every prime number ppppp, the bundle ξpξpxi_(p)\xi_{p}ξp lifts to a class in [P2p−1,Gr2]A1P2p−1,Gr2A1[P^(2p-1),Gr_(2)]_(A^(1))\left[\mathbb{P}^{2 p-1}, \mathrm{Gr}_{2}\right]_{\mathbb{A}^{1}}[P2p−1,Gr2]A1.
Remark 5.8. This is established by constructing motivic homotopy classes lifting α1α1alpha_(1)\alpha_{1}α1 and α12α12alpha_(1)^(2)\alpha_{1}^{2}α12. In our situation, the collapse map takes the form
and the lift must come from an element of [Sn−1,n,SL2]A1Sn−1,n,SL2A1[S^(n-1,n),SL_(2)]_(A^(1))\left[S^{n-1, n}, S L_{2}\right]_{\mathbb{A}^{1}}[Sn−1,n,SL2]A1. The class α1α1alpha_(1)\alpha_{1}α1 can be lifted using ideas related to those discussed in 4.1 in conjunction with a motivic version of Serre's classical ppppp-local splitting of compact Lie groups [9, THEOREM 2], the resulting lift has the wrong weight to lift to a group as above. Since the class α12α12alpha_(1)^(2)\alpha_{1}^{2}α12 is torsion, we can employ a weightshifting mechanism to fix this issue. In this direction, there are host of other vector bundles that are analogous to the Rees bundles that one might investigate from this point of view, e.g., bundles that can be built out of Toda's unstable ααalpha\alphaα-family [62]. Likewise, even the surjectivity assertion in Conjecture 5.5 is unknown for P5P5P^(5)\mathbb{P}^{5}P5.
The motivic version of the unstable Adams-Novikov resolution for BGLrBGLrBGL_(r)\mathrm{BGL}_{r}BGLr yields a spectral sequence that, under the cellularity assumption on XXXXX should converge to a (completion of) the set of rank rrrrr motivic vector bundles on XXXXX. The resulting spectral sequence can be compared to its topological counterpart and Wilson Space Hypothesis combined with the cellularity assumption on XXXXX would imply that the two spectral sequences coincide.
ACKNOWLEDGMENTS
The authors would like to acknowledge the profound influence of M. Hopkins and F. Morel on their work. Many of the results presented here would not have been possible without their insights and vision. Both authors would also like to thank all their colleagues, coauthors, and friends for their help and support.
FUNDING
Aravind Asok was partially supported by NSF Awards DMS-1802060 and DMS-2101898.
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ARAVIND ASOK
Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532, USA, asok @usc.edu
Bridgeland stability conditions and wall-crossing have provided answers to many questions in algebraic geometry a priori unrelated to derived categories, including hyperkähler varieties-their rational curves, their birational geometry, their automorphisms, and their moduli spaces-, Brill-Noether questions, Noether-Lefschetz loci, geometry of cubic fourfolds, or higher-rank Donaldson-Thomas theory. Our goal is to answer the question: why? What makes these techniques so effective, and what exactly do they add beyond, for example, classical vector bundle techniques?
The common underlying strategy can be roughly summarized as follows. For each stability condition on a derived category Db(X)Db(X)D^(b)(X)\mathrm{D}^{\mathrm{b}}(X)Db(X) of an algebraic variety XXXXX and each numerical class, moduli spaces of semistable objects in Db(X)Db(X)D^(b)(X)\mathrm{D}^{\mathrm{b}}(X)Db(X) exist as proper algebraic spaces. This formalism includes many previously studied moduli spaces: moduli spaces of Gieseker- or slope-stable sheaves, of stable pairs, or of certain equivalences classes of rational curves in XXXXX. The set of stability conditions on Db(X)Db(X)D^(b)(X)\mathrm{D}^{\mathrm{b}}(X)Db(X) has the structure of a complex manifold; when we vary the stability condition, stability of a given object only changes when we cross the walls of a well-defined wall and chamber structure.
The typical ingredients when approaching a problem with stability conditions are the following:
(large volume) There is a point in the space of stability conditions where stable objects have a "classical" interpretation, e.g. as Gieseker-stable sheaves.
(point of interest) There is a point in the space of stability conditions where stability has strong implications, e.g., vanishing properties, or even there is no semistable object of a given numerical class.
(wall-crossing) It is possible to analyze the finite set of walls between these two points, and how stability changes when crossing each wall.
In general, it is quite clear from the problem which are the points of interest, and the main difficulty consists in analyzing the walls. In the ideal situation, which leads to sharp exact results, these walls can be characterized purely numerically; there are only a few such ideal situations, K3 surfaces being one of them. Otherwise, the study of walls can get quite involved, even though there are now a number of more general results available, e.g., a wallcrossing formula for counting invariants arising from moduli spaces.
We illustrate the case of K3 surfaces, or more generally K3 categories, in Section 3, with applications to hyperkähler varieties, to Brill-Noether theory of curves, and to the geometry of special cubic fourfolds. The study of other surfaces or the higher-dimensional case becomes more technical, and the existence of Bridgeland stability conditions is not yet known in full generality. There are weaker notions of stability, which in the threefold case already lead to striking results. We give an overview of this circle of ideas in Section 4, along with three applications related to curves. We give a brief introduction to stability conditions in Section 2, and pose some questions for future research in Section 5.
Derived categories of coherent sheaves on varieties have been hugely influential in recent years; we refer to [17,19,46,74][17,19,46,74][17,19,46,74][17,19,46,74][17,19,46,74] for an overview of the theory. Moduli spaces of sheaves on K3 surfaces have largely been influenced by [61]; we refer to [63] for an overview of applications of these techniques and to [39] for the higher-dimensional case of hyperkähler manifolds. For the original motivation from physics, we refer to [26,41].
Our survey completely omits the quickly developing theory and applications of stability conditions on Kuznetsov components of Fano threefolds. We also will not touch the rich subject of extra structures on spaces of stability conditions, developed, for example, in the foundational papers [22,23]; we also refer to [72] for a symplectic perspective.
2. STABILITY CONDITIONS ON DERIVED CATEGORIES
Recall slope-stability for sheaves on an integral projective curve CCCCC : we set
μ(E)=degErkE∈(−∞,+∞], visualized by Z(E)=−degE+irkEμ(E)=degâ¡Erkâ¡E∈(−∞,+∞], visualized by Z(E)=−degâ¡E+irkEmu(E)=(deg E)/(rk E)in(-oo,+oo],quad" visualized by "Z(E)=-deg E+irkE\mu(E)=\frac{\operatorname{deg} E}{\operatorname{rk} E} \in(-\infty,+\infty], \quad \text { visualized by } Z(E)=-\operatorname{deg} E+i \mathrm{rk} Eμ(E)=degâ¡Erkâ¡E∈(−∞,+∞], visualized by Z(E)=−degâ¡E+irkE
and call a sheaf slope-semistable if every subsheaf F⊂EF⊂EF sub EF \subset EF⊂E satisfies μ(F)≤μ(F)μ(F)≤μ(F)mu(F) <= mu(F)\mu(F) \leq \mu(F)μ(F)≤μ(F). The set of semistable sheaves of fixed rank and degree is bounded and can be parameterized by a projective moduli space. Moreover, semistable sheaves generate Coh(C)Cohâ¡(C)Coh(C)\operatorname{Coh}(C)Cohâ¡(C), in the sense that every sheaf EEEEE admits a (unique and functorial) Harder-Narasimhan (HN) filtration
with El/El−1El/El−1E_(l)//E_(l-1)E_{l} / E_{l-1}El/El−1 semistable, and μ(E1)>μ(E2/E1)>⋯>μ(Em/Em−1)μE1>μE2/E1>⋯>μEm/Em−1mu(E_(1)) > mu(E_(2)//E_(1)) > cdots > mu(E_(m)//E_(m-1))\mu\left(E_{1}\right)>\mu\left(E_{2} / E_{1}\right)>\cdots>\mu\left(E_{m} / E_{m-1}\right)μ(E1)>μ(E2/E1)>⋯>μ(Em/Em−1).
How to generalize this to a variety XXXXX of dimension n≥2n≥2n >= 2n \geq 2n≥2 ? Given a polarization HHHHH, one can define the slope μHμHmu_(H)\mu_{H}μH using Hn−1⋅ch1(E)Hn−1â‹…ch1â¡(E)H^(n-1)*ch_(1)(E)H^{n-1} \cdot \operatorname{ch}_{1}(E)Hn−1â‹…ch1â¡(E) as the degree. To distinguish, e.g., the slope of the structure sheaf OXOXO_(X)\mathcal{O}_{X}OX from that of an ideal sheaf Ix⊂OXIx⊂OXI_(x)subO_(X)\mathscr{I}_{x} \subset \mathcal{O}_{X}Ix⊂OX for x∈Xx∈Xx in Xx \in Xx∈X, we can further refine the notion of slope and use lower-degree terms of the Hilbert polynomial pE(m)=χ(E(mH))pE(m)=χ(E(mH))p_(E)(m)=chi(E(mH))p_{E}(m)=\chi(E(m H))pE(m)=χ(E(mH)) as successive tie breakers; this yields Gieseker stability.
One of the key insights in Bridgeland's notion of stability conditions introduced in [20] is that instead we can, in fact, still use a notion of slope-stability, defined as the quotient of "degree" by "rank." The price we have to pay is to replace Coh(X)Cohâ¡(X)Coh(X)\operatorname{Coh}(X)Cohâ¡(X) by another abelian subcategory AAA\mathcal{A}A of the bounded derived category Db(X)Db(X)D^(b)(X)\mathrm{D}^{\mathrm{b}}(X)Db(X) of coherent sheaves on XXXXX, and to generalize the notions of "degree" and "rank" (combined into a central charge ZZZZZ as above).
To motivate the definition, consider again slope-stability for a curve CCCCC. First, for ϕ∈(0,1]ϕ∈(0,1]phi in(0,1]\phi \in(0,1]ϕ∈(0,1], let P(ϕ)⊂Coh(C)⊂Db(C)P(Ï•)⊂Cohâ¡(C)⊂Db(C)P(phi)sub Coh(C)subD^(b)(C)\mathcal{P}(\phi) \subset \operatorname{Coh}(C) \subset \mathrm{D}^{\mathrm{b}}(C)P(Ï•)⊂Cohâ¡(C)⊂Db(C) be the category of slope-semistable sheaves EEEEE with Z(E)∈R>0⋅eiπϕZ(E)∈R>0â‹…eiπϕZ(E)inR_( > 0)*e^(i pi phi)Z(E) \in \mathbb{R}_{>0} \cdot e^{i \pi \phi}Z(E)∈R>0â‹…eiπϕ, i.e., of slope μ(E)=−cot(πϕ)μ(E)=−cotâ¡(πϕ)mu(E)=-cot(pi phi)\mu(E)=-\cot (\pi \phi)μ(E)=−cotâ¡(πϕ), and let P(ϕ+n)=P(ϕ)[n]P(Ï•+n)=P(Ï•)[n]P(phi+n)=P(phi)[n]\mathcal{P}(\phi+n)=\mathcal{P}(\phi)[n]P(Ï•+n)=P(Ï•)[n] for n∈Zn∈Zn inZn \in \mathbb{Z}n∈Z be the set of semistable objects of phase ϕ+nÏ•+nphi+n\phi+nÏ•+n. Every complex E∈Db(C)E∈Db(C)E inD^(b)(C)E \in \mathrm{D}^{\mathrm{b}}(C)E∈Db(C) has a filtration into its cohomology objects Hl(E)[−l]Hl(E)[−l]H^(l)(E)[-l]\mathscr{H}^{l}(E)[-l]Hl(E)[−l]. We can combine this with the classical HN filtration of Hl(E)[−l]Hl(E)[−l]H^(l)(E)[-l]\mathscr{H}^{l}(E)[-l]Hl(E)[−l] for each lllll to obtain a finer filtration for EEEEE where every filtration quotient is semistable, i.e., an object of P(ϕ)P(Ï•)P(phi)\mathcal{P}(\phi)P(Ï•) for ϕ∈Rϕ∈Rphi inR\phi \in \mathbb{R}ϕ∈R. The properties of this structure are axiomatized by conditions (1)-(4) in Definition 2.1 below. But crucially it can always be obtained from slope-stability in an abelian category AAA\mathcal{A}A; we just have to generalize the setting A⊂Db(A)A⊂Db(A)AsubD^(b)(A)\mathcal{A} \subset \mathrm{D}^{\mathrm{b}}(\mathcal{A})A⊂Db(A) to A⊂DA⊂DAsubD\mathscr{A} \subset \mathscr{D}A⊂D being the "heart of a bounded ttt\mathrm{t}t-structure" in a triangulated category.
When combined with the remaining conditions in Definition 2.1, the main payoff are the strong deformation and wall-crossing properties of Bridgeland stability conditions. Given any small deformation of "rank" and "degree" (equivalently, of the central charge ZZZZZ ), we can accordingly adjust the abelian category AAA\mathcal{A}A (or, equivalently, the set of semistable objects PPP\mathcal{P}P ) and obtain a new stability condition. Along such a deformation, moduli spaces of semistable objects undergo very well-behaved wall-crossing transformations.
2.1. Bridgeland stability conditions
We now consider more generally an admissible subcategory DDD\mathscr{D}D of Db(X)Db(X)D^(b)(X)\mathrm{D}^{\mathrm{b}}(X)Db(X) for a smooth and proper variety XXXXX over a field kkkkk, namely a full triangulated subcategory whose inclusion D↪Db(X)D↪Db(X)D↪D^(b)(X)\mathscr{D} \hookrightarrow \mathrm{D}^{\mathrm{b}}(X)D↪Db(X) admits both a left and a right adjoint. For instance, D=Db(X)D=Db(X)D=D^(b)(X)\mathscr{D}=\mathrm{D}^{\mathrm{b}}(X)D=Db(X); otherwise we think of DDD\mathscr{D}D as a smooth and proper noncommutative variety.
We fix a finite rank free abelian group ΛΛLambda\LambdaΛ and a group homomorphism
from the Grothendieck group K0(D)K0(D)K_(0)(D)K_{0}(\mathscr{D})K0(D) of DDD\mathscr{D}D to ΛΛLambda\LambdaΛ.
Definition 2.1. A Bridgeland stability condition on DDD\mathscr{D}D with respect to (v,Λ)(v,Λ)(v,Lambda)(v, \Lambda)(v,Λ) is a pair σ=(Z,P)σ=(Z,P)sigma=(Z,P)\sigma=(Z, \mathcal{P})σ=(Z,P) where
Z:Λ→CZ:Λ→CZ:Lambda rarrCZ: \Lambda \rightarrow \mathbb{C}Z:Λ→C is a group homomorphism, called central charge, and
P=(P(ϕ))ϕ∈RP=(P(Ï•))ϕ∈RP=(P(phi))_(phi inR)\mathscr{P}=(\mathscr{P}(\phi))_{\phi \in \mathbb{R}}P=(P(Ï•))ϕ∈R is a collection of full additive subcategories P(ϕ)⊂DP(Ï•)⊂DP(phi)subD\mathscr{P}(\phi) \subset \mathscr{D}P(Ï•)⊂D satisfying the following conditions:
(1) for all nonzero E∈P(ϕ)E∈P(Ï•)E inP(phi)E \in \mathcal{P}(\phi)E∈P(Ï•), we have Z(v(E))∈R>0⋅eiπϕZ(v(E))∈R>0â‹…eiπϕZ(v(E))inR_( > 0)*e^(i pi phi)Z(v(E)) \in \mathbb{R}_{>0} \cdot e^{i \pi \phi}Z(v(E))∈R>0â‹…eiπϕ;
(2) for all ϕ∈Rϕ∈Rphi inR\phi \in \mathbb{R}ϕ∈R, we have P(ϕ+1)=P(ϕ)[1]P(Ï•+1)=P(Ï•)[1]P(phi+1)=P(phi)[1]\mathcal{P}(\phi+1)=\mathcal{P}(\phi)[1]P(Ï•+1)=P(Ï•)[1];
(3) if ϕ1>ϕ2Ï•1>Ï•2phi_(1) > phi_(2)\phi_{1}>\phi_{2}Ï•1>Ï•2 and Ej∈P(ϕj)Ej∈PÏ•jE_(j)inP(phi_(j))E_{j} \in \mathcal{P}\left(\phi_{j}\right)Ej∈P(Ï•j), then Hom(E1,E2)=0Homâ¡E1,E2=0Hom(E_(1),E_(2))=0\operatorname{Hom}\left(E_{1}, E_{2}\right)=0Homâ¡(E1,E2)=0;
(4) (Harder-Narasimhan filtrations) for all nonzero E∈DE∈DE inDE \in \mathscr{D}E∈D, there exist real numbers ϕ1>ϕ2>⋯>ϕmÏ•1>Ï•2>⋯>Ï•mphi_(1) > phi_(2) > cdots > phi_(m)\phi_{1}>\phi_{2}>\cdots>\phi_{m}Ï•1>Ï•2>⋯>Ï•m and a finite sequence of morphisms
such that the cone of slsls_(l)s_{l}sl is a non-zero object of P(ϕl)PÏ•lP(phi_(l))\mathcal{P}\left(\phi_{l}\right)P(Ï•l);
(5) (support property) there exists a quadratic form QQQQQ on ΛR=Λ⊗RΛR=Λ⊗RLambda_(R)=Lambda oxR\Lambda_{\mathbb{R}}=\Lambda \otimes \mathbb{R}ΛR=Λ⊗R such that
the kernel of ZZZZZ is negative definite with respect to QQQQQ, and
for all E∈P(ϕ)E∈P(Ï•)E inP(phi)E \in \mathcal{P}(\phi)E∈P(Ï•) for any ϕÏ•phi\phiÏ• we have Q(v(E))≥0Q(v(E))≥0Q(v(E)) >= 0Q(v(E)) \geq 0Q(v(E))≥0;
(6) (openness of stability) the property of being in P(ϕ)P(Ï•)P(phi)\mathcal{P}(\phi)P(Ï•) is open in families of objects in DDD\mathscr{D}D over any scheme;
(7) (boundedness) objects in P(ϕ)P(Ï•)P(phi)\mathcal{P}(\phi)P(Ï•) with fixed class v∈Λv∈Λv in Lambdav \in \Lambdav∈Λ are parameterized by a kkkkk-scheme of finite type.
An object of the subcategory P(ϕ)P(Ï•)P(phi)\mathcal{P}(\phi)P(Ï•) is called σσsigma\sigmaσ-semistable of phase ϕÏ•phi\phiÏ•, and σσsigma\sigmaσ-stable if it admits no non-trivial subobject in P(ϕ)P(Ï•)P(phi)\mathcal{P}(\phi)P(Ï•). The set of Bridgeland stability conditions on DDD\mathscr{D}D is denoted by Stab(D)Stabâ¡(D)Stab(D)\operatorname{Stab}(\mathscr{D})Stabâ¡(D), where we omit the dependence on (v,Λ)(v,Λ)(v,Lambda)(v, \Lambda)(v,Λ) from the notation.
Conditions (1)-(4) form the original definition in [20] and ensure we have a notion of slope-stability. The support property is necessary to show that stability conditions vary continuously (see Theorem 2.2 below) and admit a well-behaved wall and chamber structure: fundamentally, this is due to the simple linear algebra consequence that given C>0C>0C > 0C>0C>0, there are only finitely many classes w∈Λw∈Λw in Lambdaw \in \Lambdaw∈Λ of semistable objects with |Z(w)|<C|Z(w)|<C|Z(w)| < C|Z(w)|<C|Z(w)|<C. Conditions (6) and (7) were introduced in [73,74], with similar versions appearing previously in [42, sECTION 3]; they guarantee the existence of moduli spaces of semistable objects.
Theorem 2.2 (Bridgeland deformation theorem). The set Stab(D)Stabâ¡(D)Stab(D)\operatorname{Stab}(\mathscr{D})Stabâ¡(D) has the structure of a complex manifold such that the natural map
For conditions (1)-(5), this is a reformulation of Bridgeland's main result [20, THEOREM 1.2]. It says that σ=(Z,P)σ=(Z,P)sigma=(Z,P)\sigma=(Z, \mathcal{P})σ=(Z,P) can be deformed uniquely given a small deformation of Z⇝Z′Zâ‡Z′Zâ‡Z^(')Z \leadsto Z^{\prime}Zâ‡Z′, roughly as long as Z′(E)≠0Z′(E)≠0Z^(')(E)!=0Z^{\prime}(E) \neq 0Z′(E)≠0 remains true for all σσsigma\sigmaσ-semistable objects EEEEE. (More precisely, any path where QQQQQ remains negative definite on KerZ′Kerâ¡Z′Ker Z^(')\operatorname{Ker} Z^{\prime}Kerâ¡Z′ can be lifted uniquely to a path in Stab(D)Stabâ¡(D)Stab(D)\operatorname{Stab}(\mathscr{D})Stabâ¡(D).) With the additional conditions (6) and (7), Theorem 2.2 was proved in [73, THEOREM 3.20] and [67, SECTION 4.4], where the most difficult aspect is to show that openness of stability is preserved under deformations.
The theory has been developed over an arbitrary base scheme in [8]. A stability condition over a base is the datum of a stability condition on each fiber, such that families of objects over the base have locally constant central charges, satisfy openness of stability, and a global notion of HN filtration after base change to a one-dimensional scheme; moreover, we impose a global version of the support property and of boundedness. An analogue of Theorem 2.2 holds; differently to the absolute case, assuming the support property is not enough and the proof requires the additional conditions (6) and (7).
The construction of Bridgeland stability conditions is discussed in Section 4; in particular, they exist on surfaces and certain threefolds.
2.2. Stability conditions as polarizations
It was first suggested in the arXiv version of [21] to think of σσsigma\sigmaσ as a polarization of the noncommutative variety DDD\mathscr{D}D. We now review some results partly justifying this analogy. A polarization of a variety XXXXX by an ample line bundle HHHHH gives projective moduli spaces of HHHHH-Gieseker-stable sheaves; the following two results provide an analogue.
Theorem 2.3 (Toda, Alper, Halpern-Leistner, Heinloth). Given σ∈Stab(D)σ∈Stabâ¡(D)sigma in Stab(D)\sigma \in \operatorname{Stab}(\mathscr{D})σ∈Stabâ¡(D) and v∈Λv∈Λv in Lambdav \in \Lambdav∈Λ, there is a finite type Artin stack Mσ(v)Mσ(v)M_(sigma)(v)\mathcal{M}_{\sigma}(v)Mσ(v) of σσsigma\sigmaσ-semistable objects EEEEE with v(E)=vv(E)=vv(E)=vv(E)=vv(E)=v and fixed phase. In characteristic zero, it has a proper good moduli space Mσ(v)Mσ(v)M_(sigma)(v)M_{\sigma}(v)Mσ(v) in the sense of Alper.
Proof. The existence as Artin stack is [73, THEOREM 3.20], while the existence of a good moduli space is proven in [2, THEOREM 7.25]; See alSO [8, THEOREM 21.24].
Theorem 2.4. The algebraic space Mσ(v)Mσ(v)M_(sigma)(v)M_{\sigma}(v)Mσ(v) admits a Cartier divisor ℓσℓσℓ_(sigma)\ell_{\sigma}ℓσ that has strictly positive degree on every curve. In characteristic zero, if Mσ(v)Mσ(v)M_(sigma)(v)M_{\sigma}(v)Mσ(v) is smooth, or more generally if it has QQQ\mathbb{Q}Q-factorial log-terminal singularities, then Mσ(v)Mσ(v)M_(sigma)(v)M_{\sigma}(v)Mσ(v) is projective.
Proof. The existence of the Cartier divisor and its properties is the Positivity Lemma in [12], see also [8, tHEOREM 21.25]. The projectivity follows from [76, COROLLARY 3.4].
As studied extensively in Donaldson theory in the 1990s, the Gieseker-moduli spaces may change as HHHHH crosses walls in the ample cone.
Theorem 2.5. Fix a vector v∈Λv∈Λv in Lambdav \in \Lambdav∈Λ. Then there exists a locally finite union WvWvW_(v)\mathcal{W}_{v}Wv of realcodimension one submanifolds in Stab(D)Stabâ¡(D)Stab(D)\operatorname{Stab}(\mathscr{D})Stabâ¡(D), called walls, such that on every connected component CCC\mathcal{C}C of the complement Stab(D)∖WvStabâ¡(D)∖WvStab(D)\\W_(v)\operatorname{Stab}(\mathscr{D}) \backslash \mathcal{W}_{v}Stabâ¡(D)∖Wv, called a chamber, the moduli space Mσ(v)Mσ(v)M_(sigma)(v)M_{\sigma}(v)Mσ(v) is independent of the choice σ∈ℓσ∈ℓsigma inâ„“\sigma \in \mathcal{\ell}σ∈ℓ.
Theorem 2.5 follows from the results in [21, SECTION 9]; see also [73, PROPOSITION 2.8] and [10, PROPOSITION 3.3]. The set WvWvW_(v)\mathcal{W}_{v}Wv consists of stability conditions for which there exists an exact triangle A→E→BA→E→BA rarr E rarr BA \rightarrow E \rightarrow BA→E→B of semistable objects of the same phase with v(E)=vv(E)=vv(E)=vv(E)=vv(E)=v, but v(A)v(A)v(A)v(A)v(A) not proportional to vvvvv. Locally, the wall is defined by Z(A)Z(A)Z(A)Z(A)Z(A) being proportional to Z(E)Z(E)Z(E)Z(E)Z(E), and the objects EEEEE is unstable on the side where arg(Z(A))>arg(Z(E))argâ¡(Z(A))>argâ¡(Z(E))arg(Z(A)) > arg(Z(E))\arg (Z(A))>\arg (Z(E))argâ¡(Z(A))>argâ¡(Z(E)); often it is stable near the wall on the other side, e.g., when AAAAA and BBBBB are stable and the extension is nontrivial. The support property (5) is again crucial in the proof of Theorem 2.5: it constrains the classes a=v(A),b=v(B)a=v(A),b=v(B)a=v(A),b=v(B)a=v(A), b=v(B)a=v(A),b=v(B) involved in a wall, and locally that produces a finite set.
Sometimes, one can describe WvWvW_(v)\mathcal{W}_{v}Wv completely, namely when we know which of the moduli spaces Mσ(a)Mσ(a)M_(sigma)(a)M_{\sigma}(a)Mσ(a) and Mσ(b)Mσ(b)M_(sigma)(b)M_{\sigma}(b)Mσ(b) are nonempty.
2.3. K3 categories
Such descriptions of WvWvW_(v)\mathcal{W}_{v}Wv have been particularly powerful in the case of K3K3K3\mathrm{K} 3K3 categories; it has also been carried out completely for Db(P2)DbP2D^(b)(P^(2))\mathrm{D}^{\mathrm{b}}\left(\mathbb{P}^{2}\right)Db(P2), where the answer is more involved [24,54]. For this section, we work over the complex numbers and let DDD\mathscr{D}D be
(1) the derived category D=Db(S)D=Db(S)D=D^(b)(S)\mathscr{D}=\mathrm{D}^{\mathrm{b}}(S)D=Db(S) of a smooth projective K3K3K3\mathrm{K} 3K3 surface, or
of the derived category of a smooth cubic fourfold YYYYY, or
(3) the Kuznetsov component of a Gushel-Mukai fourfold defined in [47].
In (1) we can also allow a Brauer twist; one expects further examples of Kuznetsov components of Fano varieties where similar results hold. In all these cases, DDD\mathscr{D}D is a Calabi-Yau-2 category: there is a functorial isomorphism Hom(E,F)=Hom(F,E[2])∨Homâ¡(E,F)=Homâ¡(F,E[2])∨Hom(E,F)=Hom(F,E[2])^(vv)\operatorname{Hom}(E, F)=\operatorname{Hom}(F, E[2])^{\vee}Homâ¡(E,F)=Homâ¡(F,E[2])∨ for all E,F∈DE,F∈DE,F inDE, F \in \mathscr{D}E,F∈D.
Moreover, it has an associated integral weight two Hodge structure H∗(D,Z)H∗(D,Z)H^(**)(D,Z)H^{*}(\mathscr{D}, \mathbb{Z})H∗(D,Z) with an even pairing (_,_)_,_(_,_)\left(\_, \_\right)(_,_); in the case of a K3K3K3\mathrm{K} 3K3 surface, H∗(Db(S),Z)=H∗(S,Z)H∗Db(S),Z=H∗(S,Z)H^(**)(D^(b)(S),Z)=H^(**)(S,Z)H^{*}\left(\mathrm{D}^{\mathrm{b}}(S), \mathbb{Z}\right)=H^{*}(S, \mathbb{Z})H∗(Db(S),Z)=H∗(S,Z) with H0H0H^(0)H^{0}H0 and H4H4H^(4)H^{4}H4 considered to be (1,1)(1,1)(1,1)(1,1)(1,1)-classes; in the other cases, the underlying lattice is the same, and after the initial indirect construction in [1] there is now an intrinsic construction based on the topological K-theory and Hochschild homology of DDD\mathscr{D}D [64]. There is a Mukai vector v:K0(D)→H1,1(D,Z)v:K0(D)→H1,1(D,Z)v:K_(0)(D)rarrH^(1,1)(D,Z)v: K_{0}(\mathscr{D}) \rightarrow H^{1,1}(\mathscr{D}, \mathbb{Z})v:K0(D)→H1,1(D,Z) satisfying χ(E,F)=−(v(E),v(F))χ(E,F)=−(v(E),v(F))chi(E,F)=-(v(E),v(F))\chi(E, F)=-(v(E), v(F))χ(E,F)=−(v(E),v(F)) for all E,F∈DE,F∈DE,F inDE, F \in \mathscr{D}E,F∈D.
In all three cases, there is a main component Stab†(D)⊂Stab(D)Stab†â¡(D)⊂Stabâ¡(D)Stab^(†)(D)sub Stab(D)\operatorname{Stab}^{\dagger}(\mathscr{D}) \subset \operatorname{Stab}(\mathscr{D})Stab†â¡(D)⊂Stabâ¡(D) with an effective version of Theorem 2.2 for Λ=H1,1(D,Z)Λ=H1,1(D,Z)Lambda=H^(1,1)(D,Z)\Lambda=H^{1,1}(\mathscr{D}, \mathbb{Z})Λ=H1,1(D,Z) : the map ZZZ\mathbb{Z}Z is a covering of an explicitly described open subset of Hom(Λ,CHomâ¡(Λ,CHom(Lambda,C\operatorname{Hom}(\Lambda, \mathbb{C}Homâ¡(Λ,C ), see [21] for case (1), [8,9] for case (2), and [65] for case (3).
Now consider a family of such K3K3K3\mathrm{K} 3K3 categories, given by a family of K3K3K3\mathrm{K} 3K3 surfaces or Fano fourfolds over a base scheme, respectively. In this case, Mukai's classical deformation argument applies: every stable object EEEEE in a given fiber is simple, i.e., it satisfies Hom(E,E)=CHomâ¡(E,E)=CHom(E,E)=C\operatorname{Hom}(E, E)=\mathbb{C}Homâ¡(E,E)=C, and so Ext2(E,E)=CExt2â¡(E,E)=CExt^(2)(E,E)=C\operatorname{Ext}^{2}(E, E)=\mathbb{C}Ext2â¡(E,E)=C by Serre duality; therefore the obvious obstruction to extending EEEEE across the family, namely that v(E)v(E)v(E)v(E)v(E) remains a Hodge class, is the only one. Extending such deformation arguments to DDD\mathscr{D}D was the original motivation for introducing stability conditions for families of noncommutative varieties, see [8, SECTION 31]. They allows us to deduce nonemptiness of moduli spaces from the previously known case of K3K3K3\mathrm{K} 3K3 surfaces (and simplify the previous classical argument for Gieseker stability on K3 surfaces by reduction to elliptically fibered K3K3K3\mathrm{K} 3K3 s, see [18]), which leads to the following result.
Theorem 2.6 (Mukai, Huybrechts, O'Grady, Yoshioka, Toda [8,12,65][8,12,65][8,12,65][8,12,65][8,12,65] ). Let v∈H1,1(D,Z)v∈H1,1(D,Z)v inH^(1,1)(D,Z)v \in H^{1,1}(\mathscr{D}, \mathbb{Z})v∈H1,1(D,Z) be primitive, and σ∈Stab†(D)σ∈Stab†â¡(D)sigma inStab^(†)(D)\sigma \in \operatorname{Stab}^{\dagger}(\mathcal{D})σ∈Stab†â¡(D) be generic. Then Mσ(v)Mσ(v)M_(sigma)(v)M_{\sigma}(v)Mσ(v) is nonempty if and only if v2:=(v,v)≥−2v2:=(v,v)≥−2v^(2):=(v,v) >= -2v^{2}:=(v, v) \geq-2v2:=(v,v)≥−2; in this case, it is a smooth projective irreducible holomorphic symplectic (IHS) variety.
More precisely, Mσ(v)Mσ(v)M_(sigma)(v)M_{\sigma}(v)Mσ(v) is of K[n]K[n]K^([n])\mathrm{K}^{[n]}K[n]-type, where n=(v2+2)/2n=v2+2/2n=(v^(2)+2)//2n=\left(v^{2}+2\right) / 2n=(v2+2)/2, i.e., it is deformation equivalent to the Hilbert scheme of nnnnn points on a K3 surface (see [25,34] for the basic theory of irreducible holomorphic symplectic varieties). If v2≥2v2≥2v^(2) >= 2v^{2} \geq 2v2≥2, the Mukai morphism
induced by a (quasi)universal family gives an identification of H2(Mσ(v),Z)H2Mσ(v),ZH^(2)(M_(sigma)(v),Z)H^{2}\left(M_{\sigma}(v), \mathbb{Z}\right)H2(Mσ(v),Z) with v⊥v⊥v^(_|_)v^{\perp}v⊥. If v2=0v2=0v^(2)=0v^{2}=0v2=0, then Mσ(v)Mσ(v)M_(sigma)(v)M_{\sigma}(v)Mσ(v) is a K3K3K3\mathrm{K} 3K3 surface and H2(Mσ(v),Z)H2Mσ(v),ZH^(2)(M_(sigma)(v),Z)H^{2}\left(M_{\sigma}(v), \mathbb{Z}\right)H2(Mσ(v),Z) is identified with v⊥/vv⊥/vv^(_|_)//vv^{\perp} / vv⊥/v.
Knowing exactly which semistable objects exist then allows us to describe exactly when we are on a wall. While a complete result as in [11, THEOREM 5.7] also needs to treat essential aspects of the wall-crossing behavior, the basic result is simple to state:
Theorem 2.7 ([11]). Let v∈H1,1(D,Z)v∈H1,1(D,Z)v inH^(1,1)(D,Z)v \in H^{1,1}(\mathscr{D}, \mathbb{Z})v∈H1,1(D,Z) be a primitive class. Then σ=(Z,P)∈Stab†(D)σ=(Z,P)∈Stab†â¡(D)sigma=(Z,P)inStab^(†)(D)\sigma=(Z, \mathcal{P}) \in \operatorname{Stab}^{\dagger}(\mathscr{D})σ=(Z,P)∈Stab†â¡(D) lies on a wall for vvvvv if and only if there exists classes a,b∈H1,1(D,Z)a,b∈H1,1(D,Z)a,b inH^(1,1)(D,Z)a, b \in H^{1,1}(\mathscr{D}, \mathbb{Z})a,b∈H1,1(D,Z) with v=a+bv=a+bv=a+bv=a+bv=a+b, a2,b2≥−2a2,b2≥−2a^(2),b^(2) >= -2a^{2}, b^{2} \geq-2a2,b2≥−2 and Z(a),Z(b)Z(a),Z(b)Z(a),Z(b)Z(a), Z(b)Z(a),Z(b) are positive real multiples of Z(v)Z(v)Z(v)Z(v)Z(v).
And the fundamental reason is similarly simple to explain: by Theorem 2.6, this allows for the existence of extensions
(2.1)0→A→E→B→0(2.1)0→A→E→B→0{:(2.1)0rarr A rarr E rarr B rarr0:}\begin{equation*}
0 \rightarrow A \rightarrow E \rightarrow B \rightarrow 0 \tag{2.1}
\end{equation*}(2.1)0→A→E→B→0
where v(A)=a,v(E)=v,v(B)=bv(A)=a,v(E)=v,v(B)=bv(A)=a,v(E)=v,v(B)=bv(A)=a, v(E)=v, v(B)=bv(A)=a,v(E)=v,v(B)=b, and A,E,BA,E,BA,E,BA, E, BA,E,B are all semistable of the same phase For stronger results, we need to know when such EEEEE can become stable near the wall.
3. CONSTRUCTIONS BASED ON K3 CATEGORIES
In this section we present three applications of stability conditions on K3K3K3\mathrm{K} 3K3 categories, to irreducible holomorphic symplectic varieties, to curves, and to cubic fourfolds.
3.1. Curves in irreducible holomorphic symplectic manifolds
Let MMMMM be a smooth projective irreducible holomorphic symplectic (IHS) variety of K3[n]K3[n]K3^([n])\mathrm{K3}^{[n]}K3[n]-type, with n≥2n≥2n >= 2n \geq 2n≥2. We let qMqMq_(M)q_{M}qM be the Beauville-Bogomolov-Fujiki quadratic form on H2(M,Z)H2(M,Z)H^(2)(M,Z)H^{2}(M, \mathbb{Z})H2(M,Z). By [25, sEction 3.7.1], there exists a canonical extension
of lattices and weight-2 Hodge structures, where the lattice Λ~MΛ~Mwidetilde(Lambda)_(M)\widetilde{\Lambda}_{M}Λ~M is isometric to the extended K3K3K3\mathrm{K} 3K3 lattice U⊕4⊕E8(−1)⊕2U⊕4⊕E8(−1)⊕2U^(o+4)o+E_(8)(-1)^(o+2)U^{\oplus 4} \oplus E_{8}(-1)^{\oplus 2}U⊕4⊕E8(−1)⊕2. Let us denote by v∈Λ~Mv∈Λ~Mv in widetilde(Lambda)_(M)v \in \widetilde{\Lambda}_{M}v∈Λ~M a generator of ϑ(H2(M,Z))⊥Ï‘H2(M,Z)⊥vartheta(H^(2)(M,Z))^(_|_)\vartheta\left(H^{2}(M, \mathbb{Z})\right)^{\perp}Ï‘(H2(M,Z))⊥ : it is of type (1,1)(1,1)(1,1)(1,1)(1,1) and square v2=2n−2v2=2n−2v^(2)=2n-2v^{2}=2 n-2v2=2n−2. The lattice Λ~MΛ~Mwidetilde(Lambda)_(M)\widetilde{\Lambda}_{M}Λ~M is called the Markman-Mukai lattice associated to MMMMM. If M=Mσ(v)M=Mσ(v)M=M_(sigma)(v)M=M_{\sigma}(v)M=Mσ(v), for a stability condition σ∈Stab†(Db(S))σ∈Stab†â¡Db(S)sigma inStab^(†)(D^(b)(S))\sigma \in \operatorname{Stab}^{\dagger}\left(\mathrm{D}^{\mathrm{b}}(S)\right)σ∈Stab†â¡(Db(S)) on a K3K3K3\mathrm{K} 3K3 surface SSSSS, then Λ~M=H∗(S,Z)Λ~M=H∗(S,Z)widetilde(Lambda)_(M)=H^(**)(S,Z)\widetilde{\Lambda}_{M}=H^{*}(S, \mathbb{Z})Λ~M=H∗(S,Z) with the Mukai pairing, the notation for the vector vvvvv is coherent, and ϑMÏ‘Mvartheta_(M)\vartheta_{M}Ï‘M is the Mukai morphism mentioned after Theorem 2.6.
We let Pos(M)Posâ¡(M)Pos(M)\operatorname{Pos}(M)Posâ¡(M) be the connected component of the positive cone of MMMMM containing an ample divisor class:
The following result rephrases and proves a conjecture by Hassett-Tschinkel and gives a complete description of the ample cone of MMMMM.
Theorem 3.1. Let MMMMM be a smooth projective IHS variety of K[n]K[n]K^([n])\mathrm{K}^{[n]}K[n]-type. The ample cone of MMMMM is a connected component of
Theorem 3.1 is proved in [11] for moduli spaces of stable sheaves/complexes on a K3K3K3\mathrm{K} 3K3 surface, and extended in [7] to all IHS of K3[n]K3[n]K3^([n])\mathrm{K} 3{ }^{[n]}K3[n]-type, by using deformation theory of rational curves on IHS varieties.
The approach to Theorem 3.1 via wall-crossing is as follows. Let SSSSS be a K3 surface and M=Mσ0(v)M=Mσ0(v)M=M_(sigma_(0))(v)M=M_{\sigma_{0}}(v)M=Mσ0(v) be a moduli space of σ0σ0sigma_(0)\sigma_{0}σ0-stable objects in Db(S)Db(S)D^(b)(S)\mathrm{D}^{\mathrm{b}}(S)Db(S), where v∈v∈v inv \inv∈H1,1(Db(S),Z)H1,1Db(S),ZH^(1,1)(D^(b)(S),Z)H^{1,1}\left(\mathrm{D}^{\mathrm{b}}(S), \mathbb{Z}\right)H1,1(Db(S),Z) is a primitive vector of square v2≥2v2≥2v^(2) >= 2v^{2} \geq 2v2≥2. As σσsigma\sigmaσ varies in the chamber φφvarphi\varphiφ containing σ0σ0sigma_(0)\sigma_{0}σ0, Theorem 2.4 gives a family of ample divisor classes ℓσℓσℓ_(sigma)\ell_{\sigma}ℓσ in Pos(M)Posâ¡(M)Pos(M)\operatorname{Pos}(M)Posâ¡(M). When σσsigma\sigmaσ reaches a wall of ℓâ„“â„“\mathcal{\ell}â„“, as given by Theorem 2.7, the class ℓσℓσℓ_(sigma)\ell_{\sigma}ℓσ remains nef. On the other hand, consider an object EEEEE that becomes strictly semistable on the wall, admitting an exact
sequence as in (2.1). Varying the extension class in a line in P(Ext1(B,A))PExt1â¡(B,A)P(Ext^(1)(B,A))\mathbb{P}\left(\operatorname{Ext}^{1}(B, A)\right)P(Ext1â¡(B,A)) produces a P1P1P^(1)\mathbb{P}^{1}P1 of such objects, and Theorem 2.4 shows that ℓσℓσℓ_(sigma)\ell_{\sigma}ℓσ has degree zero on this curve. We have found an extremal curve and, dually, a boundary wall of the ample cone.
FIGURE 1
The approach to Theorem 3.1.
We summarize the history underlying Theorem 3.1 with the diagram in Figure 1. The analogue of Theorem 2.6 for Gieseker-stable sheaves involves a two-step argument, using autoequivalences and deformations. Wall-crossing techniques then imply the existence of Bridgeland stable objects on K3 surfaces, and thus Theorem 2.6. As discussed above, a finer wall-crossing analysis based on Theorem 2.7 then produces the extremal rational curves on moduli spaces that appear implicitly as extremal curves in Theorem 3.1. Finally, another deformation argument, involving rational curves, deduces Theorem 3.1 for all IHS manifolds of K3[n]K3[n]K3^([n])\mathrm{K} 3{ }^{[n]}K3[n]-type. Wall-crossing combined with stability conditions in families can also simplify the approach to Theorem 2.6, see [18].
3.2. Curves
Consider a Brill-Noether (BN) wall in Stab(Db(X))Stabâ¡Db(X)Stab(D^(b)(X))\operatorname{Stab}\left(\mathrm{D}^{\mathrm{b}}(X)\right)Stabâ¡(Db(X)) for a variety XXXXX : the structure sheaf OXOXO_(X)\mathcal{O}_{X}OX is stable and of the same phase ϕÏ•phi\phiÏ• as objects EEEEE of a fixed class vvvvv. Then OXOXO_(X)\mathcal{O}_{X}OX is an object of the abelian category P(ϕ)P(Ï•)P(phi)\mathcal{P}(\phi)P(Ï•) with no subobjects; hence the evaluation map OX⊗OX⊗O_(X)ox\mathcal{O}_{X} \otimesOX⊗H0(E)→EH0(E)→EH^(0)(E)rarr EH^{0}(E) \rightarrow EH0(E)→E must be injective, giving a short exact sequence
(3.1)0→OX⊗H0(E)→E→Q→0(3.1)0→OX⊗H0(E)→E→Q→0{:(3.1)0rarrO_(X)oxH^(0)(E)rarr E rarr Q rarr0:}\begin{equation*}
0 \rightarrow \mathcal{O}_{X} \otimes H^{0}(E) \rightarrow E \rightarrow Q \rightarrow 0 \tag{3.1}
\end{equation*}(3.1)0→OX⊗H0(E)→E→Q→0
where Q∈P(ϕ)Q∈P(Ï•)Q inP(phi)Q \in \mathscr{P}(\phi)Q∈P(Ï•) is also semistable. Applying known inequalities for Chern classes of semistable objects to ch(Q)=v−h0(E)ch(OX)châ¡(Q)=v−h0(E)châ¡OXch(Q)=v-h^(0)(E)ch(O_(X))\operatorname{ch}(Q)=v-h^{0}(E) \operatorname{ch}\left(\mathcal{O}_{X}\right)châ¡(Q)=v−h0(E)châ¡(OX) can directly lead to bounds on h0(E)h0(E)h^(0)(E)h^{0}(E)h0(E).
This simple idea turns out to be powerful. For a K3K3K3\mathrm{K} 3K3 surface SSSSS, we can be more precise: applying Theorem 2.6 to the class of QQQQQ, we can construct all EEEEE with given r=h0r=h0r=h^(0)r=h^{0}r=h0 as a Grassmannian bundle Gr(r,Ext1(Q,OS))Grâ¡r,Ext1â¡Q,OSGr(r,Ext^(1)(Q,O_(S)))\operatorname{Gr}\left(r, \operatorname{Ext}^{1}\left(Q, \mathcal{O}_{S}\right)\right)Grâ¡(r,Ext1â¡(Q,OS)) of extensions over the moduli space of such QQQQQ.
Corollary 3.2. Let SSSSS be a K3K3K3K 3K3 surface, v∈H1,1(S,Z)v∈H1,1(S,Z)v inH^(1,1)(S,Z)v \in H^{1,1}(S, \mathbb{Z})v∈H1,1(S,Z) primitive and σσsigma\sigmaσ be a stability condition near the Brill-Noether wall for vvvvv. If the lattice generated by vvvvv and v(OS)vOSv(O_(S))v\left(\mathcal{O}_{S}\right)v(OS) is saturated, then the locus of objects E∈Mσ(v)E∈Mσ(v)E inM_(sigma)(v)E \in M_{\sigma}(v)E∈Mσ(v) with h0(E)=rh0(E)=rh^(0)(E)=rh^{0}(E)=rh0(E)=r has expected dimension.
In [5], this is applied, in the case where Pic(S)=Z⋅HPicâ¡(S)=Zâ‹…HPic(S)=Z*H\operatorname{Pic}(S)=\mathbb{Z} \cdot HPicâ¡(S)=Zâ‹…H, to rank zero classes of the form v=(0,H,s)v=(0,H,s)v=(0,H,s)v=(0, H, s)v=(0,H,s). In this case, there are no walls between the BNBNBN\mathrm{BN}BN wall and the large volume limit; hence Corollary 3.2 applies in the large volume limit, and thus to zero-dimensional torsion sheaves supported on curves in the primitive linear system. This gives a variant of Lazarsfeld's proof [49] of the Brill-Noether theorem: every curve in the primitive linear system is Brill-Noether general.
This approach has been significantly strengthened in [27]: instead of requiring EEEEE to be semistable near the Brill-Noether wall, it is sufficient to control the classes occurring in its HN filtration. A bound on h0h0h^(0)h^{0}h0 is obtained by applying Corollary 3.2 to all HN filtration factors. Thus we need to consider a point near the Brill-Noether walls for all HNHNHN\mathrm{HN}HN factors, and which is the limit point where Z(OX)∼0ZOX∼0Z(O_(X))∼0Z\left(\mathcal{O}_{X}\right) \sim 0Z(OX)∼0.
Proposition 3.3 ([27, PRoposition 3.4]). Let SSSSS be a K3 surface of Picard rank one. There exists a limit point σ¯ÏƒÂ¯bar(sigma)\bar{\sigma}σ¯ of the space of stability conditions, with central charge Z¯Z¯bar(Z)\bar{Z}Z¯, and a constant CCCCC, such that for (most) objects in the heart, we have
h0(E)+h1(E)≤C⋅∑l|Z¯(El/El−1)|h0(E)+h1(E)≤C⋅∑l Z¯El/El−1h^(0)(E)+h^(1)(E) <= C*sum_(l)|( bar(Z))(E_(l)//E_(l-1))|h^{0}(E)+h^{1}(E) \leq C \cdot \sum_{l}\left|\bar{Z}\left(E_{l} / E_{l-1}\right)\right|h0(E)+h1(E)≤C⋅∑l|Z¯(El/El−1)|
where E0⊂E1⊂⋯⊂EmE0⊂E1⊂⋯⊂EmE_(0)subE_(1)sub cdots subE_(m)E_{0} \subset E_{1} \subset \cdots \subset E_{m}E0⊂E1⊂⋯⊂Em is the HNHNHNH NHN filtration of EEEEE near σ¯ÏƒÂ¯bar(sigma)\bar{\sigma}σ¯.
The following application completes a program originally proposed by Mukai [62]:
Theorem 3.4 ([27, 28]). Let SSSSS be a polarised K3K3K3K 3K3 surface with Pic(S)=Z⋅HPicâ¡(S)=Zâ‹…HPic(S)=Z*H\operatorname{Pic}(S)=\mathbb{Z} \cdot HPicâ¡(S)=Zâ‹…H and genus g≥11g≥11g >= 11g \geq 11g≥11, and let C∈|H|C∈|H|C in|H|C \in|H|C∈|H|. Then SSSSS is the unique K3 surface containing CCCCC, and can be reconstructed as a Fourier-Mukai partner of a Brill-Noether locus of stable vector bundles on CCCCC with prescribed number of sections.
The structure of the argument is as follows. The numerics are chosen such that there is a two-dimensional moduli space S^S^hat(S)\hat{S}S^, necessarily a K3 surface, of stable bundles EEEEE on SSSSS whose restriction E|CECE|_(C)\left.E\right|_{C}E|C is automatically in the Brill-Noether locus. Conversely, given a stable bundle VVVVV on CCCCC, its push-forward i∗Vi∗Vi_(**)Vi_{*} Vi∗V along i:C↪Si:C↪Si:C↪Si: C \hookrightarrow Si:C↪S is stable at the large volume limit. Standard wall-crossing arguments bound its HN filtration near the limit point σ¯ÏƒÂ¯bar(sigma)\bar{\sigma}σ¯ in Proposition 3.3, which then gives a bound on h0(V)h0(V)h^(0)(V)h^{0}(V)h0(V). The argument also shows that equality-the Brill-Noether condition-only holds for the HNHNHN\mathrm{HN}HN filtration E→E|C=i∗V→E(−H)[1]E→EC=i∗V→E(−H)[1]E rarr E|_(C)=i_(**)V rarr E(-H)[1]\left.E \rightarrow E\right|_{C}=i_{*} V \rightarrow E(-H)[1]E→E|C=i∗V→E(−H)[1], i.e., when VVVVV is the restriction of a vector bundle in S^S^hat(S)\hat{S}S^. Thus S^S^hat(S)\hat{S}S^ is a Brill-Noether locus on CCCCC, and SSSSS can be reconstructed as a Fourier-Mukai partner of S^S^hat(S)\hat{S}S^.
3.3. Surfaces in cubic fourfolds
Let Y⊂P5Y⊂P5Y subP^(5)Y \subset \mathbb{P}^{5}Y⊂P5 be a complex smooth cubic fourfold and let hhhhh be the class of a hyperplane section. Following [38], we say that YYYYY is special of discriminant ddddd, and write Y∈CdY∈CdY inC_(d)Y \in \mathscr{C}_{d}Y∈Cd, if there exists a surface Σ⊂YΣ⊂YSigma sub Y\Sigma \subset YΣ⊂Y such that h2h2h^(2)h^{2}h2 and ΣΣSigma\SigmaΣ span a saturated rank two lattice in H4(Y,Z)H4(Y,Z)H^(4)(Y,Z)H^{4}(Y, \mathbb{Z})H4(Y,Z) with